#493754

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1799
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1087 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^c=di,f^c=fi,g^c=ghi,g^f=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, 400950);