C₂×#12234 #484433

order	512
nilpotency class	4
isoclinism	isoclinic class of #12096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12234

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1219 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1219 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);(3,4)(5,7)(6,9)(8,10)(11,12,16,19)(13,20,22,25)(14,18,15,23)(17,24,26,21);(5,10)(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(4,7)(5,10)(8,9)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391629);