#72256691

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

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

At least 8947 conjugacy classes of subgroups; only the 31993 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=j,d^2=hi,e^2=i,f^2=j,b^a=bj,d^a=di,d^b=dh,d^c=dh,e^b=eij,e^c=eij,f^d=fi,g^e=gj

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256702
maximal normal subgroup of	#72256702

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, 1, F.10);;
gap> SetPower(coll, 3, F.10);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);