#6729699

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=gi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6636895);