Finite Group Databasebeta

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Cayley diagram

C2×#23588 #6432074

order512
nilpotency class3
isoclinismisoclinic class of #2002
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#23588

Subgroups

subgroupG/H
centerC22×C4#120
commutator, Frattini, cosocleC22×C4C25
socleC23C2×#120
prime radicalC2×#2006C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3556 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)(5,8)(6,7)(9,10)(11,12)(13,16)(14,17)(15,19)(18,23)(20,24)(21,26)(22,25);(4,7)(8,10)(11,13,15,21)(12,16,19,26)(14,22,23,20)(17,25,18,24);(12,17)(14,23)(16,24)(18,19)(20,22)(25,26);(3,5,6,9)(4,8,7,10)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6432083, C2×#6453874, C2×#6453870
maximal normal subgroup ofC2×#6432083, C2×#6453874, C2×#6453870
maximal quotient ofC2×#241504, #72309293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339270);