#7181256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(8,16)(11,17)(15,20)(18,19)(21,22,27,35,26,24,25,30)(23,29,36,28,31,34,32,33);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,31)(22,28,24,33)(25,36,27,32)(29,30,34,35);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23)(22,29)(24,34)(25,32)(26,31)(27,36)(28,35)(30,33);(7,14)(10,13)(15,20)(18,19)(21,24,27,30,26,22,25,35)(23,29,36,28,31,34,32,33);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,24)(23,31)(30,35)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088452);