C₂×#12423 #490426

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12423

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 498 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) = 512

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

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397622);