#381237

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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 1304 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=egi,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, 288433);