C₂×#15199 #608490

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

C₂×#15199

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

5110 conjugacy classes of subgroups; only the 1546 normal subgroups are shown.

normal subgroups only

This group has 5110 conjugacy classes of subgroups — too many to draw. Only its 1546 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) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(4,6)(8,9)(13,17)(16,18)(19,20,22,24)(21,25,26,23);(7,8,10,9)(12,15)(16,18);(3,4,5,6)(7,9,10,8)(11,12)(13,16)(14,15)(17,18);(11,13,14,17)(12,16,15,18)(19,21,22,26)(20,23,24,25)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515686);