C₂×#10216 #232857

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#10216

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1270 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1270 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)(6,9)(8,10)(11,12,17,22)(13,23,26,37)(14,27,30,35)(15,21,16,20)(18,31,33,25)(19,28,36,29)(24,32,40,34)(38,41,42,39);(4,7)(5,10)(8,9)(11,13)(12,18)(14,24)(15,31)(16,25)(17,26)(19,34)(20,37)(21,23)(22,33)(27,38)(28,41)(29,39)(30,40)(32,36)(35,42);(3,4,6,9)(5,8,10,7)(11,14,17,30)(12,19,22,36)(13,24,26,40)(15,29,16,28)(18,32,33,34)(20,27,21,35)(23,38,37,42)(25,41,31,39)

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, 140053);