#499560

order	512
nilpotency class	4
isoclinism	isoclinic class of #499029
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#482
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2101
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

479 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)(3,15,18,43)(4,10,21,35)(5,11)(6,14)(7,12)(8,13)(9,17,33,29)(16,45)(19,39,50,63)(20,52)(22,36,30,44)(23,37,55,62)(24,38)(25,42)(26,57,56,32)(27,40,46,64)(28,41)(31,49)(34,51)(47,59)(48,58)(53,60)(54,61);(1,3,8,29,6,18,5,17)(2,9,11,15,14,33,13,43)(4,20,30,51,21,53,22,54)(7,26,28,19,25,56,24,50)(10,34,36,52,35,61,44,60)(12,39,38,57,42,63,41,32)(16,46,49,55,47,27,48,23)(31,40,45,62,58,64,59,37);(1,4,5,22,6,21,8,30)(2,10,11,36,14,35,13,44)(3,16,17,48,18,47,29,49)(7,27,24,55,25,46,28,23)(9,31,15,45,33,58,43,59)(12,40,38,62,42,64,41,37)(19,51,26,20,50,54,56,53)(32,60,39,34,57,52,63,61);(4,21)(10,35)(16,47)(20,53)(22,30)(23,55)(27,46)(31,58)(34,61)(36,44)(37,62)(40,64)(45,59)(48,49)(51,54)(52,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406756);