C₂×#17301 #845186

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

direct product

C₂×#17301

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

2626 conjugacy classes of subgroups; only the 786 normal subgroups are shown.

normal subgroups only

This group has 2626 conjugacy classes of subgroups — too many to draw. Only its 786 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)(8,10)(11,12,14,16)(13,15,17,18)(19,20)(21,24)(22,23)(25,26)(28,31)(32,33);(4,7)(5,9)(13,17)(15,18)(20,22)(24,26)(27,28,30,31)(29,32,34,33);(3,4,6,7)(5,8,9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,25)(20,24,22,26)(27,28)(29,33)(30,31)(32,34);(3,5)(4,8)(6,9)(7,10)(11,13)(12,15)(14,17)(16,18)(19,22,23,20)(21,24,25,26)(27,29,30,34)(28,32,31,33)

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, 752382);