C₂×#12088 #483834

order	512
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12088

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

446 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 42
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4,7,5,6,8,10,9)(11,12,15,19,17,21,16,32)(13,22,25,40,23,41,26,37)(14,27,30,39,29,38,31,20)(18,33,34,24,35,28,36,42);(4,8)(5,9)(11,13,17,23)(12,18,21,35)(14,28,29,33)(15,26,16,25)(19,36,32,34)(20,37,39,40)(22,27,41,38)(24,31,42,30);(3,5,7,8,6,9,10,4)(12,19,21,32)(13,23)(14,29)(18,34,35,36)(20,38,39,27)(22,37,41,40)(25,26)(30,31)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#484041, C₂×#484098, C₂×#484139, …
maximal normal subgroup of	C₂×#484041, C₂×#484098, C₂×#484139, …
maximal quotient of	C₂×#140100

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391030);