#72308559

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#674425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×#127 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5685 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|a^2=g,c^2=ghj,d^2=f,e^2=i,i^2=j,c^a=cj,c^b=cgh,d^a=df,d^b=df,e^a=ei,e^c=ej,e^d=eh,i^a=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308619, #72308921
maximal normal subgroup of	#72308619, #72308921

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