#507243

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,30)(20,26)(21,37)(22,28)(23,29)(24,27)(25,32)(31,40)(33,45)(34,46)(35,42)(36,44)(38,41)(39,43)(47,48);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,31,38,21,35,39,36,22,32,23,33,24,20,34,47)(18,25,40,45,37,26,43,48,28,30,29,41,27,42,46,44);(10,13)(14,15)(17,20,22,35)(18,26,28,42)(19,21,32,24)(23,38,31,33)(25,27,30,37)(29,41,40,45)(34,47,39,36)(43,44,46,48);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,24)(18,27,28,37)(19,20,32,35)(23,34,31,39)(25,26,30,42)(29,43,40,46)(33,36,38,47)(41,44,45,48)

Polycyclic presentation:

2^9|b^2=efg,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414439);