#148297

order	512
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11095
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 55493);