#499915

order	512
nilpotency class	4
isoclinism	isoclinic class of #499914
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 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1346 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)(3,15)(4,20)(5,14)(6,27)(7,11)(8,13)(9,17)(10,22)(12,30)(16,46)(18,42)(19,39)(21,41)(23,54)(24,55)(25,34)(26,43)(28,36)(29,33)(31,40)(32,61)(35,52)(37,58)(38,56)(44,48)(45,51)(47,53)(49,64)(50,63)(57,62)(59,60);(1,3)(2,9)(4,16)(5,25)(6,18)(7,17)(8,19)(10,35)(11,39)(12,33)(13,34)(14,15)(20,53)(21,50)(22,48)(23,47)(24,49)(26,59)(27,45)(28,52)(29,31)(30,51)(32,37)(36,62)(38,44)(40,60)(41,46)(42,43)(54,64)(55,63)(56,61)(57,58);(1,4,8,24)(2,10,13,37)(3,16,19,49)(5,23,7,21)(6,28,31,56)(9,32,34,35)(11,38,14,36)(12,41,43,54)(15,44,39,62)(17,50,25,47)(18,52,29,61)(20,40,55,27)(22,26,58,30)(33,64,42,46)(45,63,60,53)(48,59,57,51);(1,3)(2,9)(4,16)(5,25)(6,29)(7,17)(8,19)(10,32)(11,39)(12,42)(13,34)(14,15)(18,31)(20,53)(21,50)(22,57)(23,47)(24,49)(26,51)(27,60)(28,61)(30,59)(33,43)(35,37)(36,44)(38,62)(40,45)(41,46)(48,58)(52,56)(54,64)(55,63)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407111);