#391683

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

896 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(25,26,29,31,28,30,27,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,21)(23,24)(26,30)(31,32);(1,4,5,3)(2,6,7,8)(10,14)(11,15)(18,21)(22,24)(25,27,28,29)(26,31,30,32);(3,4)(6,8)(11,15)(13,16)(17,19,20,23)(18,22,21,24)(25,27,28,29)(26,31,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=fg,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,d^c=dei

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, 298879);