C₂×#29657 #6434714

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×#29657

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3738 conjugacy classes of subgroups; only the 626 normal subgroups are shown.

normal subgroups only

This group has 3738 conjugacy classes of subgroups — too many to draw. Only its 626 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) = 64

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341910);