#6698976

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	#35821	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1833 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)(11,15)(12,14)(13,16)(17,18,22,20)(19,27,30,24)(21,26,23,25)(28,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,20,27)(21,29,23,31)(25,32,26,28);(10,14)(13,15)(17,20,22,18)(19,28,30,32)(21,26,23,25)(24,31,27,29);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,25,20,26)(19,29,30,31)(24,32,27,28);(2,5)(6,8)(11,16)(13,15)(17,21)(18,26)(19,30)(20,25)(22,23)(24,27)

Polycyclic presentation:

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

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, 6606172);