C₂×#29656 #6434713

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂×#29656

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4838 conjugacy classes of subgroups; only the 626 normal subgroups are shown.

normal subgroups only

This group has 4838 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) = 64

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#29656⋊C₃
maximal subgroup of	#72268310, C₂×#29656⋊C₃
maximal normal subgroup of	#72268310, C₂×#29656⋊C₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341909);