Finite Group Databasebeta

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Cayley diagram

C2×#18261 #6346557

order512
nilpotency class3
isoclinismisoclinic class of #18252
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18261

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6346652, C2×#6346766, C2×#6346732
maximal normal subgroup ofC2×#6346652, C2×#6346766, C2×#6346732
maximal quotient of#72309336, #72309338

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253753);