C₂×#10516 #238803

order	512
nilpotency class	3
isoclinism	isoclinic class of #10512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10516

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

930 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 194 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)(5,8)(6,9)(7,10)(11,12,16,20)(13,21,24,36)(14,25,28,18)(15,29,31,19)(17,30,33,23)(22,38,26,37)(27,35,42,40)(32,39,34,41);(3,4)(5,9)(6,8)(7,10)(11,13,16,24)(12,17,20,33)(14,26,28,22)(15,30,31,23)(18,34,25,32)(19,36,29,21)(27,39,42,41)(35,37,40,38);(4,8)(5,6)(9,10)(11,14,15,27,16,28,31,42)(12,18,19,35,20,25,29,40)(13,22,23,39,24,26,30,41)(17,32,21,37,33,34,36,38)

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, 145999);