#499907

order	512
nilpotency class	4
isoclinism	isoclinic class of #499905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23918
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1330 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)(3,15,21,35)(4,22,26,42)(5,13,28,11)(6,29,31,45)(8,20,36,18)(9,37,41,24)(10,25,43,23)(12,44,46,27)(16,47,51,61)(17,33,49,59)(19,34,56,62)(30,39,57,40)(32,52,60,54)(38,64,63,50)(48,53,58,55);(1,3)(2,8)(4,17)(5,20)(6,19)(7,21)(9,38)(10,33)(11,35)(12,34)(13,15)(14,36)(16,25)(18,28)(22,47)(23,51)(24,52)(26,49)(27,55)(29,58)(30,50)(31,56)(32,39)(37,54)(40,60)(41,63)(42,61)(43,59)(44,53)(45,48)(46,62)(57,64);(1,4)(2,9)(3,16)(5,23)(6,30)(7,26)(8,32)(10,12)(11,39)(13,40)(14,41)(15,38)(17,20)(18,49)(19,54)(21,51)(22,29)(24,27)(25,28)(31,57)(33,48)(34,47)(35,63)(36,60)(37,44)(42,45)(43,46)(50,55)(52,56)(53,64)(58,59)(61,62);(1,4)(2,10)(3,17)(5,25)(6,24)(7,26)(8,33)(9,12)(11,42)(13,22)(14,43)(15,47)(16,20)(18,51)(19,52)(21,49)(23,28)(27,30)(29,40)(31,37)(32,48)(34,38)(35,61)(36,59)(39,45)(41,46)(44,57)(50,55)(53,64)(54,56)(58,60)(62,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407103);