C₂×#9991 #232573

order	512
nilpotency class	3
isoclinism	isoclinic class of #9986
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#9991

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2150 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2150 conjugacy classes of subgroups — too many to draw. Only its 218 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)(13,23,27,34)(14,28,32,37)(15,21,16,20)(18,26,35,25)(19,30,38,31)(24,33,40,41)(29,42,39,36);(7,10)(11,13,17,27)(12,18,22,35)(14,29,32,39)(15,26,16,25)(19,36,38,42)(20,34,21,23)(24,31,40,30)(28,41,37,33);(3,5)(4,7)(6,9)(8,10)(11,14,17,32)(12,19,22,38)(13,24,27,40)(15,30,16,31)(18,33,35,41)(20,37,21,28)(23,36,34,42)(25,29,26,39)

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, 139769);