Finite Group Databasebeta

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Cayley diagram

C2×#1378 #12552

order256
nilpotency class3
isoclinismisoclinic class of #99
automorphismAut(G) order 4096
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24

Aliases

direct productC2×#1378

Subgroups

subgroupG/H
centerC2×C8C22⋊C4
commutatorC22C2×C4×C8
Frattini, cosocleC22×C8C23
socleC22#336
prime radicalC24C2×C8
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C23×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

293 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);(3,4,6,10,7,12,17,24,8,11,18,23,19,25,30,33)(5,13,15,26,16,27,29,34,14,9,20,21,28,22,31,32);(3,5)(4,9,11,13)(6,15)(7,16)(8,14)(10,21,23,26)(12,22,25,27)(17,29)(18,20)(19,28)(24,32,33,34)(30,31)

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#1378, #146310, #146312, …
maximal normal subgroup ofC4×#1378, #146310, #146312, …
maximal quotient ofC2×#7140, #126557, #126568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5540);