Finite Group Databasebeta

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Cayley diagram

C2×#25748 #6443304

order512
nilpotency class3
isoclinismisoclinic class of #24464
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#25748

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72312619, #72312788, C2×#6443305, …
maximal normal subgroup of#72312619, #72312788, C2×#6443305, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350500);