#497278

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×C₂²⋊C₈
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1039 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404474);