Finite Group Databasebeta

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Cayley diagram

C2×#8317 #184907

order512
nilpotency class3
isoclinismisoclinic class of #2021
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#8317

Subgroups

subgroupG/H
centerC22×C4#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23#379
prime radicalC2×#3459C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2760 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);(3,4)(5,9,10,7)(6,8)(11,12,14,17,15,18,21,24)(13,16,20,23,19,22,25,26);(3,5)(4,7)(6,10)(8,9)(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26);(3,4)(5,7)(6,8)(9,10)(13,19)(16,22)(20,25)(23,26)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#184919, C2×#208820, C2×#208822
maximal normal subgroup ofC2×#184919, C2×#208820, C2×#208822

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92103);