C₂×#12429 #490451

order	512
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12429

Subgroups

	subgroup	G/H
center, socle	C₂³	#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 556 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)(5,8)(9,10)(11,12)(13,21)(14,17)(15,27)(16,30)(18,23)(19,29)(20,26)(22,37)(24,41)(25,38)(28,36)(31,35)(32,39)(33,40)(34,42);(3,4,8,10,6,9,5,7)(12,17)(18,32)(19,33)(20,30)(21,37)(27,38)(34,36)(35,41);(11,13,14,22)(12,18,17,32)(15,28,25,42)(16,31,26,24)(19,35,33,41)(20,36,30,34)(21,38,37,27)(23,40,39,29)

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, 397647);