Finite Group Databasebeta

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Cayley diagram

C2×#10924 #354065

order512
nilpotency class3
isoclinismisoclinic class of #468
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#10924

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC24C2×C22⋊C4
prime radicalC23×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

3354 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);(4,6)(7,8,9,10)(11,12)(13,19)(14,22)(15,18)(16,21)(17,23)(20,25)(24,26);(11,13)(12,16,18,24)(14,21)(15,20)(17,19,22,25)(23,26);(3,4,5,6)(8,10)(13,20)(16,24)(19,25)(21,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239302, C2×#432837, #72258592, …
maximal normal subgroup of#72239302, C2×#432837, #72258592, …
maximal quotient of#72239292, #72239293, #72267413

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261261);