#381163

order	512
nilpotency class	3
isoclinism	isoclinic class of #195981
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1090 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=gi,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,d^c=de,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, 288359);