#497281

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

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×C₂²⋊C₈
prime radical	#3584	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404477);