C₂×#17308 #845226

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

C₂×#17308

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

3098 conjugacy classes of subgroups; only the 826 normal subgroups are shown.

normal subgroups only

This group has 3098 conjugacy classes of subgroups — too many to draw. Only its 826 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);(4,7)(5,9)(11,12,14,16)(13,17,18,15)(19,20,23,24)(21,25,26,22)(29,33)(31,34);(4,7)(5,9)(13,18)(15,17)(20,24)(22,25)(27,28,30,32)(29,34,33,31);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,25)(22,24)(23,26)(27,29,30,33)(28,31,32,34);(3,5)(4,8)(6,9)(7,10)(11,13,14,18)(12,15,16,17)(19,22,23,25)(20,26,24,21)(27,28)(29,31)(30,32)(33,34)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752422);