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Cayley diagram

C2×#401 #2400

order128
nilpotency class2
isoclinismisoclinic class of #392
automorphismAut(G) order 688128
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#401

Subgroups

subgroupG/H
center, socle, prime radicalC24C23
commutator, Frattini, cosocleC23C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×#401⋊C3, C2×#401⋊C7
maximal subgroup ofC4×#401, #14251, #14333, …
maximal normal subgroup ofC4×#401, #14251, #14333, …
maximal quotient ofC2×#1443, C4×#401, C22⋊#401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1132);