C₂×#10936 #354029

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#10936

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4046 conjugacy classes of subgroups; only the 778 normal subgroups are shown.

normal subgroups only

This group has 4046 conjugacy classes of subgroups — too many to draw. Only its 778 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);(4,6)(11,12)(13,19,21,16)(14,17)(15,18)(20,23,26,25)(22,24);(7,8,9,10)(11,13)(12,16)(14,20)(15,21)(17,23)(18,19)(22,26)(24,25);(3,4,5,6)(11,14,15,22)(12,17,18,24)(13,20,21,26)(16,23,19,25)

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, 261225);