C₂×#8753 #190129

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#8753

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1718 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 258 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)(5,9,10,7)(6,8)(11,12,15,20,16,19,29,34)(13,21,23,35,24,17,28,31)(14,25,27,39,26,18,32,33)(22,30,38,41,37,36,40,42);(11,13,16,24)(12,17,19,21)(14,22,26,37)(15,28,29,23)(18,30,25,36)(20,35,34,31)(27,40,32,38)(33,42,39,41);(3,5)(4,7)(6,10)(8,9)(11,14)(12,18,19,25)(13,22)(15,27)(16,26)(17,30,21,36)(20,33,34,39)(23,38)(24,37)(28,40)(29,32)(31,41,35,42)

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, 97325);