#509295

order	512
nilpotency class	4
isoclinism	isoclinic class of #509294
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30567
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28183	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2955 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

521 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416491);