#382372

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator, socle	C₂⁴	C₂³×C₄
prime radical	C₂⁴×D₄	C₄
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5794 conjugacy classes of subgroups; only the 506 normal subgroups are shown.

normal subgroups only

This group has 5794 conjugacy classes of subgroups — too many to draw. Only its 506 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,4)(9,10)(11,14)(12,13)(15,16);(5,6)(7,8)(10,13)(14,16);(2,4)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18,19,22,20,21,23,24);(11,15)(14,16)(18,21)(22,24)

Polycyclic presentation:

2^9|c^2=h,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=dei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289568);