C₂×#12094 #484154

order	512
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#12094

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1603 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1603 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391350);