#501082

order	512
nilpotency class	4
isoclinism	isoclinic class of #501080
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24378
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60471	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1032 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408278);