C₂×#9982 #232507

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#9982

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2214 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 2214 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128

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

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

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, 139703);