#127133

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#7928
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,12,7,15,28,43,8,14,6,13,27,42,29,44)(3,16,18,45,20,31,52,58,21,47,19,46,51,60,53,59)(4,22,23,41,24,37,54,40,25,11,26,38,55,10,35,36)(9,30,32,56,34,57,61,17,33,48,39,49,62,64,63,50);(1,3,6,19,7,20,29,53,8,21,5,18,27,51,28,52)(2,9,13,32,15,34,44,61,14,33,12,39,42,62,43,63)(4,17,23,48,24,49,54,64,25,50,26,30,55,56,35,57)(10,31,36,59,22,47,41,45,37,60,40,58,11,16,38,46);(1,4)(2,10,14,37)(3,17)(5,26)(6,23)(7,24)(8,25)(9,31,33,60)(11,15,22,42)(12,40,13,36)(16,34,47,62)(18,30)(19,48)(20,49)(21,50)(27,55)(28,35)(29,54)(32,59,39,58)(38,44,41,43)(45,63,46,61)(51,56)(52,57)(53,64)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ch,d^a=dg,g^a=gi,e^b=eg,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34329);