C₂×#12518 #491464

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

direct product

C₂×#12518

Subgroups

	subgroup	G/H
center, socle	C₂³	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 218 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)(14,26)(15,18)(16,29)(17,23)(19,22)(20,27)(24,33)(25,40)(28,41)(30,39)(31,42)(32,37)(34,36)(35,38);(11,13,22,36,16,25,28,39,15,24,37,42,27,38,14,23)(12,17,26,35,20,31,32,33,18,30,41,40,29,34,19,21);(3,4,8,10,6,9,5,7)(12,18)(17,30)(19,32)(20,29)(21,33)(26,41)(31,34)(35,40)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398660);