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Cayley diagram

C2×#1413 #13597

order256
nilpotency class5
isoclinismisoclinic class of #1412
automorphismAut(G) = #72318502 (order 4096) · Inn(G) = #354 · Out(G) = #534
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×#1413

Subgroups

subgroupG/H
center, socleC22#354
commutatorC42C22×C4
Frattini, cosocleC4⋊Q8C23
prime radicalC23#99
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ G
LECSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#94785, C4×#1413, #151936, …
maximal normal subgroup of#94785, C4×#1413, #151936, …
maximal quotient ofC2×#7358, C2×#7367, #149702, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6585);