#6687414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#42810	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308226, #72308331
maximal normal subgroup of	#72308226, #72308331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594610);