#493785

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1802
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1051 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fh,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, 400981);