C₂×#12764 #493580

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#12764

Subgroups

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

2452 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2452 conjugacy classes of subgroups — too many to draw. Only its 322 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) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2);(3,4)(5,6)(7,8)(9,17,20,32)(10,14)(11,24,26,15)(12,16)(13,19)(18,35,21,34)(22,33,38,30)(23,31)(25,28)(27,36)(29,37);(7,9,12,20)(8,13,16,28)(10,21,23,18)(11,25,26,19)(14,29,31,27)(15,32,24,17)(22,36,38,37)(30,35,33,34);(4,6)(7,10,12,23)(8,14,16,31)(9,18,20,21)(11,22,26,38)(13,27,28,29)(15,33,24,30)(17,34,32,35)(19,36,25,37)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400776);