Finite Group Databasebeta

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Cayley diagram

C2×#18167 #6346421

order512
nilpotency class3
isoclinismisoclinic class of #18162
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#18167

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2388C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4090 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);(7,10)(11,12)(13,20)(14,18)(15,24)(16,19)(17,22)(21,25)(23,26);(3,4)(5,7)(6,8)(9,10)(11,13,16,23)(12,17,19,25)(14,22,15,21)(18,26,24,20);(3,5)(4,7)(6,9)(8,10);(11,14,16,15)(12,18,19,24)(13,21,23,22)(17,20,25,26)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6346597, C2×#6346423, C2×#6346601
maximal normal subgroup ofC2×#6346597, C2×#6346423, C2×#6346601

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253617);