C₂³×#362 #7622509

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂³×#362

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#102
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4002 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.

normal subgroups only

This group has 4002 conjugacy classes of subgroups — too many to draw. Only its 1346 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 38 points:

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

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, 7529705);