Finite Group Databasebeta

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Cayley diagram

C2×#20924 #6371403

order512
nilpotency class3
isoclinismisoclinic class of #447
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C211

Aliases

direct productC2×#20924

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC24C22×D4
prime radicalC23×#135C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

6674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6371408, C2×#6437509, C2×#6437496
maximal normal subgroup ofC2×#6371408, C2×#6437509, C2×#6437496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278599);