Finite Group Databasebeta

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Cayley diagram

C2×#25593 #6449086

order512
nilpotency class3
isoclinismisoclinic class of #24468
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25593

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

4002 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)(5,9)(6,7)(8,10)(14,22)(16,24)(18,23)(21,26);(4,7)(8,9)(11,12)(13,19)(14,23)(15,17)(16,21)(18,22)(20,25)(24,26);(11,13)(12,16,17,24)(14,21)(15,20)(18,19,23,25)(22,26);(3,5,6,10)(4,8,7,9)(13,20)(16,24)(19,25)(21,26)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72319796, #72320018, #72320049
maximal normal subgroup of#72319796, #72320018, #72320049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356282);