C₂³×#491 #10575229

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂³×#491

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

7986 conjugacy classes of subgroups; only the 3746 normal subgroups are shown.

normal subgroups only

This group has 7986 conjugacy classes of subgroups — too many to draw. Only its 3746 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) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 30 points:

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

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, 10482425);