#499562

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2101
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320632, #72320647, #72320763
maximal normal subgroup of	#72320632, #72320647, #72320763

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406758);