#6690646

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

1920 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,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	#72308624, #72308849
maximal normal subgroup of	#72308624, #72308849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597842);