Finite Group Databasebeta

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Cayley diagram

C2×#18332 #6346880

order512
nilpotency class3
isoclinismisoclinic class of #18316
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#18332

Subgroups

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

4216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340417, #72340418, C2×#485614
maximal normal subgroup of#72340417, #72340418, C2×#485614

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254076);