#512491

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419687);