#72297209

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

85817 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

(1,2)(3,4)(6,9)(10,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,24)(25,27)(26,28);(2,4)(7,11)(10,12)(13,15)(14,19)(16,17)(18,24)(20,27)(21,28)(22,23)(25,26);(6,9)(10,12)(15,24)(16,26)(19,23)(20,28);(2,4)(7,11)(10,12)(13,16,18,26)(14,20,22,28)(15,25,24,17)(19,21,23,27);(2,4)(5,6)(7,12)(8,9)(10,11)(13,16)(14,20)(15,17)(18,26)(19,27)(21,23)(22,28)(24,25);(13,17)(14,21)(15,26)(16,24)(18,25)(19,20)(22,27)(23,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,27)(18,22)(21,25)(23,24)(26,28)

Polycyclic presentation:

2^10|d^2=h,b^a=bj,c^b=ch,d^a=dhj,d^b=dh,d^c=dh,e^a=ei,e^c=eij,e^d=eh,f^b=fh,f^d=fh,f^e=fh,g^b=gi,g^d=gi,g^e=gi,g^f=gh

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72297308, #72297583, #72297794, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);