C₂×#13360 #498912

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#13360

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2161 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 2161 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312323, #72312327, C₂×#503225, …
maximal normal subgroup of	#72312323, #72312327, C₂×#503225, …
maximal quotient of	#72313437

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406108);