#7181135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,26,30)(23,27,34,40)(24,29,35,44)(25,28,37,31)(32,39,45,51)(33,38,47,41)(36,42,48,43)(46,49,52,50);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,34)(22,27,30,40)(24,32,35,45)(25,33,37,47)(28,38,31,41)(29,39,44,51)(36,46,48,52)(42,49,43,50);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(22,28)(25,37)(27,38)(29,42)(30,31)(33,47)(36,48)(39,49)(40,41)(43,44)(46,52)(50,51);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24,26,35)(22,29,30,44)(23,32,34,45)(25,36,37,48)(27,39,40,51)(28,42,31,43)(33,46,47,52)(38,49,41,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,30)(24,35)(27,40)(28,31)(32,45)(36,48)(38,41)(46,52)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088331);