C₂×#12335 #489168

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12335

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

794 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 162 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);(4,7)(5,9)(8,10)(11,12,17,21)(13,22,25,41)(14,19,16,31)(15,27,30,39)(18,33,36,24)(20,26,38,29)(23,40,32,37)(28,34,42,35);(5,9)(7,10)(11,13,17,25)(12,18,21,36)(14,23,16,32)(15,28,30,42)(19,34,31,35)(20,37,38,40)(22,39,41,27)(24,26,33,29);(3,4,6,8)(5,10,9,7)(11,14,17,16)(12,19,21,31)(13,23,25,32)(15,26,30,29)(18,34,36,35)(20,27,38,39)(22,40,41,37)(24,28,33,42)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396364);