#497830

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×C₄⋊C₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

626 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=ci,d^b=de,e^b=ei,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, 405026);