#6690864

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309046
maximal normal subgroup of	#72309046

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6598060);