#72298401

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

35365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 20 points:

(2,3)(5,6)(7,8)(9,10)(11,12)(14,15)(17,18)(19,20);(2,3)(6,7)(9,10,12,11)(14,15)(17,18,19,20);(1,2,4,3)(5,6,8,7)(9,10)(11,12)(17,18)(19,20);(10,11)(13,14,16,15)(17,18,19,20);(1,3,4,2)(5,7,8,6)(9,11,12,10)(13,15,16,14)(17,18)(19,20)

Polycyclic presentation:

2^10|b^2=g,c^2=fh,d^2=hj,e^2=fgj,b^a=bf,c^a=cfh,c^b=cfgh,d^a=dgi,d^b=dghj,d^c=dg,e^a=efgj,e^b=efj,e^c=eghij,e^d=eg

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72298586, #72298594, #72298623
maximal normal subgroup of	#72298586, #72298594, #72298623

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