C₂×#9976 #232435

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#9976

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

2202 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 2202 conjugacy classes of subgroups — too many to draw. Only its 210 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)(11,12,17,21)(13,22,26,34)(14,27,31,37)(15,32,16,20)(18,25,35,24)(19,29,38,30)(23,40,41,33)(28,39,42,36);(7,10)(11,13)(12,18)(14,28)(15,25)(16,24)(17,26)(19,36)(20,34)(21,35)(22,32)(23,29)(27,33)(30,41)(31,42)(37,40)(38,39);(3,5)(4,7)(6,9)(8,10)(11,14,17,31)(12,19,21,38)(13,23,26,41)(15,29,16,30)(18,33,35,40)(20,37,32,27)(22,39,34,36)(24,28,25,42)

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, 139631);