#72302996

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

At least 8660 conjugacy classes of subgroups; only the 31837 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)(7,9)(8,10)(11,12)(13,14,19,22)(15,23,26,20)(16,21,18,17)(24,25,27,28);(5,6,8,10)(7,11,12,9)(13,15)(14,20)(16,27)(17,28)(18,24)(19,26)(21,25)(22,23);(2,4)(7,12)(9,11);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,19,18)(14,21,22,17)(15,24,26,27)(20,28,23,25);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,24)(17,25)(18,27)(19,26)(21,28)(22,23);(13,17)(14,16)(15,25)(18,22)(19,21)(20,24)(23,27)(26,28);(13,18,19,16)(14,17,22,21)(15,24,26,27)(20,28,23,25)

Polycyclic presentation:

2^10|a^2=j,b^2=hj,d^2=j,g^2=j,b^a=bh,d^b=dh,d^c=di,e^a=ej,e^b=ehj,e^c=ehj,f^b=fj,g^e=gj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303075
maximal normal subgroup of	#72303075

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