C₂×#17054 #612044

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

Aliases

direct product

C₂×#17054

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 842 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 842 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)(8,10)(11,12)(13,15)(14,17)(16,18)(19,20,23,24)(21,25,26,22)(28,29)(30,34);(4,7)(8,10)(12,13)(16,17)(20,24)(21,26)(27,28,31,29)(30,32,34,33);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,16,18,17)(19,21)(20,25)(22,24)(23,26)(27,29,31,28)(30,32,34,33);(3,5,6,9)(4,8,7,10)(11,14,15,18)(12,16,13,17)(19,22,23,25)(20,26,24,21)(27,30)(28,32)(29,33)(31,34)

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, 519240);