C₂×#29658 #6434715

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#29658

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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3500 conjugacy classes of subgroups; only the 618 normal subgroups are shown.

normal subgroups only

This group has 3500 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) = 64

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

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

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, 6341911);