#148556

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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)(3,16,21,46)(4,23,27,42)(5,13)(6,12)(7,33)(8,34)(9,15)(10,18,38,30)(11,26,43,31)(14,32)(17,36)(19,39,20,48)(22,45,35,37)(24,56)(25,44,58,55)(28,60,59,41)(29,47)(40,50)(49,53)(51,64)(52,54)(57,63)(61,62);(1,3)(2,10)(4,24)(5,20)(6,30)(7,18)(8,35)(9,21)(11,40)(12,45)(13,46)(14,48)(15,38)(16,47)(17,26)(19,29)(22,32)(23,54)(25,57)(27,53)(28,61)(31,52)(33,37)(34,39)(36,42)(41,63)(43,62)(44,49)(50,59)(51,58)(55,56)(60,64);(1,4,5,25,8,28,6,31,9,27,29,58,32,59,7,26)(2,11,12,41,14,44,13,42,15,43,33,60,34,55,47,23)(3,17,18,50,22,51,19,53,21,52,30,61,35,57,20,24)(10,36,16,49,39,63,37,40,38,54,46,56,48,64,45,62)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ei,g^c=gi

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, 55752);