#508540

order	512
nilpotency class	4
isoclinism	isoclinic class of #508540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29925
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

456 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:

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

Polycyclic presentation:

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

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, 415736);