#410875

order	512
nilpotency class	3
isoclinism	isoclinic class of #394043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2613
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15092
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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:

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

Polycyclic presentation:

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

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, 318071);