#6808866

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1895 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)(23,28)(24,29)(25,27)(26,30)(31,34)(32,35)(33,36);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(22,27)(23,24)(26,33)(28,35)(29,34)(31,32);(2,4)(21,23,26,24,25,31,33,32)(22,28,30,29,27,34,36,35);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,27)(23,31)(24,32)(30,36);(7,15)(10,13)(14,20)(18,19)(21,24,33,23,25,32,26,31)(22,29,36,28,27,35,30,34)

Polycyclic presentation:

2^9|a^2=fi,c^2=h,d^2=fi,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.

Relationships

maximal subgroup of	#72257053, #72257127, #72257204
maximal normal subgroup of	#72257053, #72257127, #72257204
maximal quotient of	#72264756

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716062);