#7186502

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 7093698);