C₂×#12698 #493368

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#12698

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3422 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 3422 conjugacy classes of subgroups — too many to draw. Only its 338 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)(18,33,21,35)(22,37,38,30)(23,31)(25,28)(27,34)(29,36);(7,9,12,20)(8,13,16,28)(10,21,23,18)(11,25,26,19)(14,29,31,27)(15,32,24,17)(22,34,38,36)(30,33,37,35);(4,6)(7,10,12,23)(8,14,16,31)(9,18,20,21)(11,22,26,38)(13,27,28,29)(15,30,24,37)(17,33,32,35)(19,34,25,36)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#493371
maximal normal subgroup of	C₂×#493371
maximal quotient of	C₂×#152128, C₂×#133078, C₂×#134066, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400564);