C₂×#29710 #6435392

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

Aliases

direct product

C₂×#29710

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3459	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3546 conjugacy classes of subgroups; only the 626 normal subgroups are shown.

normal subgroups only

This group has 3546 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) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(4,5)(8,9)(11,12)(13,16)(14,17)(15,18)(19,24)(20,23)(21,25)(22,26);(3,4)(5,7)(6,8)(9,10)(11,13)(12,16)(14,19)(15,20)(17,24)(18,23)(21,26)(22,25);(3,5,7,4)(6,8,10,9)(11,14,15,21)(12,17,18,25)(13,19,20,26)(16,22,23,24);(3,6,7,10)(4,8,5,9)

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, 6342588);