C₂×#33299 #6630850

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

direct product

C₂×#33299

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 1810 conjugacy classes of subgroups — too many to draw. Only its 738 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) = 34
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538046);