Finite Group Databasebeta

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Cayley diagram

C2×#28137 #6459454

order512
nilpotency class3
isoclinismisoclinic class of #24283
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#28137

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3003C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3220 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)(6,8)(7,10)(11,12)(13,21)(14,19)(15,18)(16,20)(17,23)(22,25)(24,26);(3,5,6,10)(4,7,8,9)(11,13,15,22,14,23,16,24)(12,17,18,26,19,21,20,25);(12,18)(13,22)(15,16)(17,21)(19,20)(23,24);(4,8)(7,9)(12,19)(13,23)(18,20)(22,24)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72216151
maximal normal subgroup of#72216151
maximal quotient ofC2×#378855

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366650);