#499509

order	512
nilpotency class	4
isoclinism	isoclinic class of #498802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23697
prime radical	#13312	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72217290, #72224960, …
maximal normal subgroup of	#72217287, #72217290, #72224960, …
maximal quotient of	C₂×#499509

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406705);