C₂×#33298 #6630849

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

direct product

C₂×#33298

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

1938 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 1938 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)(21,29)(22,25)(23,33)(24,28)(26,31)(27,30)(32,34);(4,7)(8,10)(19,21,30,32,24,31,23,22)(20,25,33,26,28,34,27,29);(11,13,14,18)(12,15,16,17)(21,31)(22,32)(25,34)(26,29);(3,5,6,9)(4,8,7,10)(19,22,23,31,24,32,30,21)(20,26,27,25,28,29,33,34)

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, 6538045);