#507263

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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 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)(19,25)(20,26)(21,36)(22,28)(23,27)(24,29)(30,38)(31,40)(32,39)(33,41)(34,42)(35,43)(37,46)(44,47)(45,48);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,30,44,21,20,34,45,24,33,22,32,23,31,37,35)(18,25,38,47,36,26,42,48,29,41,28,39,27,40,46,43);(10,13)(14,15)(17,20,24,31)(18,26,29,40)(19,23,33,21)(22,32,30,44)(25,27,41,36)(28,39,38,47)(34,35,37,45)(42,43,46,48);(2,6)(5,7)(9,11,12,16)(10,14,13,15)(17,21,24,23)(18,27,29,36)(19,31,33,20)(22,37,30,34)(25,26,41,40)(28,42,38,46)(32,45,44,35)(39,43,47,48)

Polycyclic presentation:

2^9|b^2=efgi,c^2=i,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,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, 414459);