#382448

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

3682 conjugacy classes of subgroups; only the 506 normal subgroups are shown.

normal subgroups only

This group has 3682 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) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,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, 289644);