#6808875

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	#39137	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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)(21,22,27,26)(23,28,32,35)(24,29,34,25)(30,33,36,31);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,32)(22,28,26,35)(24,33,34,31)(25,30,29,36);(2,4)(21,24,22,29,27,34,26,25)(23,30,35,31,32,36,28,33);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,34)(25,29)(30,36)(31,33);(7,15)(10,13)(14,20)(18,19)(21,25,26,34,27,29,22,24)(23,31,28,30,32,33,35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=hi,d^2=f,e^2=h,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	#72257048, #72257130, #72257204
maximal normal subgroup of	#72257048, #72257130, #72257204
maximal quotient of	#72264567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716071);