#6808856

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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×#3180	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,32)(23,33,24,29)(25,35,28,30)(26,31,34,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,24)(22,29,32,33)(25,34,28,26)(30,36,35,31);(2,4)(21,24,28,26,27,23,25,34)(22,29,30,31,32,33,35,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,28)(22,30,32,35)(23,34,24,26)(29,36,33,31);(7,15)(10,13)(14,20)(18,19)(21,26,25,24,27,34,28,23)(22,31,35,29,32,36,30,33)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6716052);