C₂×#12109 #484207

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12109

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1131 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1131 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

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

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391403);