C₂×#19096 #6351716

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#19096

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2930 conjugacy classes of subgroups; only the 570 normal subgroups are shown.

normal subgroups only

This group has 2930 conjugacy classes of subgroups — too many to draw. Only its 570 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) = 34
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

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

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, 6258912);