Finite Group Databasebeta

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Cayley diagram

C2×#24321 #6445424

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24321

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3342 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);(3,4,7,10)(5,9,6,8)(11,12,16,20)(13,17,23,26)(14,18,15,19)(21,24,22,25);(3,5,7,6)(4,8,10,9)(11,13)(12,17)(14,22)(15,21)(16,23)(18,25)(19,24)(20,26);(12,18)(13,21)(14,15)(19,20)(22,23)(24,25);(3,6,7,5)(4,9,10,8)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6445430, C2×#6453882, C2×#6453870
maximal normal subgroup ofC2×#6445430, C2×#6453882, C2×#6453870
maximal quotient ofC2×#265723, #72309298

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352620);