Finite Group Databasebeta

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Cayley diagram

C2×#18136 #6346376

order512
nilpotency class3
isoclinismisoclinic class of #18132
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18136

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268099, #72268101, C2×#11715585, …
maximal normal subgroup of#72268099, #72268101, C2×#6346377, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253572);