C₂×#10528 #238927

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10528

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4)(5,8)(6,9)(7,10)(11,12,17,21)(13,22,26,37)(14,27,30,19)(15,32,16,20)(18,25,34,24)(23,38,40,35)(28,33,39,42)(29,36,31,41);(3,4)(5,9)(6,8)(7,10)(11,13,17,26)(12,18,21,34)(14,28,30,39)(15,25,16,24)(19,35,27,38)(20,37,32,22)(23,29,40,31)(33,41,42,36);(4,8)(5,6)(9,10)(11,14,15,29,17,30,16,31)(12,19,20,36,21,27,32,41)(13,23,24,39,26,40,25,28)(18,33,22,35,34,42,37,38)

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, 146123);