C₂×#17081 #612340

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#17081

Subgroups

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

2878 conjugacy classes of subgroups; only the 914 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 914 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);(5,9)(7,10)(11,12,14,16)(13,17,18,15)(20,23)(21,25)(27,28)(29,31)(30,32)(33,34);(19,20)(21,24)(22,23)(25,26)(27,28,30,32)(29,31,34,33);(3,4,6,8)(5,7,9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,26)(22,23)(24,25)(27,28)(29,33)(30,32)(31,34);(3,5,6,9)(4,7,8,10)(11,13,14,18)(12,15,16,17)(19,21,22,25)(20,24,23,26)(27,29)(28,31)(30,34)(32,33)

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, 519536);