C₂×#29659 #6434716

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

Aliases

direct product

C₂×#29659

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3100 conjugacy classes of subgroups; only the 618 normal subgroups are shown.

normal subgroups only

This group has 3100 conjugacy classes of subgroups — too many to draw. Only its 618 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);(3,4,7,10)(5,11,13,8)(6,9,14,16)(12,17,18,15)(19,20)(21,25)(22,26)(23,27)(24,28)(29,34)(30,32)(31,33);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21)(20,25)(22,29)(23,30)(24,31)(26,34)(27,32)(28,33);(3,6)(4,9)(5,12)(7,14)(8,15)(10,16)(11,17)(13,18)(19,22,24,23)(20,26,28,27)(21,29,31,30)(25,32,33,34);(6,14)(9,16)(12,18)(15,17)(19,23,24,22)(20,27,28,26)(21,30,31,29)(25,32,33,34)

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, 6341912);