#499595

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23758
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=eh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=ghi,h^c=hi

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, 406791);