C₂×#16248 #610713

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

direct product

C₂×#16248

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3922 conjugacy classes of subgroups; only the 938 normal subgroups are shown.

normal subgroups only

This group has 3922 conjugacy classes of subgroups — too many to draw. Only its 938 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) = 512

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

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#16248⋊C₃
maximal subgroup of	#72267509, C₂×#16248⋊C₃, #72267684, …
maximal normal subgroup of	#72267509, C₂×#16248⋊C₃, #72267684, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 517909);