#72278377

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7401 conjugacy classes of subgroups; only the 1027 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=fi,c^2=gi,d^2=fi,e^2=fghj,b^a=bf,c^b=cfg,d^a=dfi,d^c=dh,e^a=efhij,e^b=efgh,e^d=egi

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278966
maximal normal subgroup of	#72278966
maximal quotient of	#72299924

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