Finite Group Databasebeta

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Cayley diagram

C2×#23865 #6437508

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C211

Aliases

direct productC2×#23865

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

6320 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);(3,4)(5,6)(11,12,17,21)(13,22,14,19)(15,20,16,25)(18,23,26,24);(7,8)(9,10)(11,13,17,14)(12,18,21,26)(15,24,16,23)(19,25,22,20);(4,6)(8,10)(11,14,17,13)(12,19,21,22)(15,24,16,23)(18,25,26,20);(11,15)(13,23)(14,24)(16,17)(18,26)(20,25)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6437517, C2×#6437509, C2×#6437519
maximal normal subgroup ofC2×#6437517, C2×#6437509, C2×#6437519

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344704);