C₂×#12196 #484791

order	512
nilpotency class	4
isoclinism	isoclinic class of #12152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12196

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂×#2006	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1149 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1149 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 128

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

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391987);