#382691

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1882 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1882 conjugacy classes of subgroups — too many to draw. Only its 282 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,29,37)(20,24,32,40)(21,33,34,25)(27,35,30,38)(28,42,43,36)(31,45,46,39)(41,48,44,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,38)(25,36)(26,37)(27,32)(31,44)(33,42)(34,43)(35,40)(39,48)(41,46)(45,47);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(17,20)(18,24)(19,27,29,30)(21,31)(22,32)(23,35,37,38)(25,39)(26,40)(28,41,43,44)(33,45)(34,46)(36,47,42,48);(3,7)(6,8)(17,21,22,34)(18,25,26,33)(19,28,29,43)(20,31,32,46)(23,36,37,42)(24,39,40,45)(27,41,30,44)(35,47,38,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,h^b=hi,d^c=de,g^c=gi

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, 289887);