C₂×#19743 #6357386

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

direct product

C₂×#19743

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2374 conjugacy classes of subgroups; only the 594 normal subgroups are shown.

normal subgroups only

This group has 2374 conjugacy classes of subgroups — too many to draw. Only its 594 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);(4,8)(6,9)(10,14)(11,15)(13,16)(17,18)(21,27)(22,28)(24,30)(26,31)(29,33)(32,34);(3,4,9,15,7,11,6,8)(5,10,16,18,12,17,13,14)(19,20)(21,24)(22,26)(23,25)(27,30)(28,31)(29,32)(33,34);(3,5)(4,10)(6,13)(7,12)(8,14)(9,16)(11,17)(15,18)(19,21,28,33,23,29,22,27)(20,24,31,34,25,32,26,30);(4,11)(5,12)(8,15)(13,16)(20,25)(24,32)(26,31)(30,34)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264582);