C₂×#28259 #6465475

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#28259

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3342 conjugacy classes of subgroups; only the 698 normal subgroups are shown.

normal subgroups only

This group has 3342 conjugacy classes of subgroups — too many to draw. Only its 698 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) = 256

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265663, C₂×#7614034, #72265690, …
maximal normal subgroup of	#72265663, C₂×#7614034, #72265690, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372671);