#496376

order	512
nilpotency class	4
isoclinism	isoclinic class of #496375
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21810
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60648	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1027 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)(3,15,18,9)(4,20)(5,26,28,11)(6,30)(7,34)(8,12)(10,23)(13,17)(14,32)(16,49,51,40)(19,38,27,48)(21,57,53,54)(22,58,59,39)(24,62)(25,56)(29,43,63,37)(31,46)(33,55)(35,45)(36,50,47,60)(41,61)(42,64)(44,52);(1,3,7,19,8,18,32,27)(2,9,34,48,12,15,14,38)(4,21,24,51,25,53,35,16)(5,17,29,33,28,52,63,6)(10,39,46,47,41,58,42,36)(11,13,37,55,26,44,43,30)(20,54,62,49,56,57,45,40)(22,31,60,61,59,64,50,23);(1,4,8,25)(2,10,12,41)(3,16,18,51)(5,22,28,59)(6,31,33,64)(7,35,32,24)(9,36,15,47)(11,40,26,49)(13,45,44,62)(14,46,34,42)(17,23,52,61)(19,53,27,21)(20,55,56,30)(29,50,63,60)(37,57,43,54)(38,39,48,58);(1,5)(2,11)(3,17)(4,22)(6,27)(7,29)(8,28)(9,13)(10,40)(12,26)(14,43)(15,44)(16,23)(18,52)(19,33)(20,39)(21,31)(24,60)(25,59)(30,38)(32,63)(34,37)(35,50)(36,45)(41,49)(42,57)(46,54)(47,62)(48,55)(51,61)(53,64)(56,58)

Polycyclic presentation:

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

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, 403572);