#509831

order	512
nilpotency class	4
isoclinism	isoclinic class of #509829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31032
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24297	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417027);