Finite Group Databasebeta

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Cayley diagram

C2×#25825 #6445791

order512
nilpotency class3
isoclinismisoclinic class of #24412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#25825

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2990 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)(13,21,24,18)(14,17,19,15)(22,26,25,23);(3,5,7,6)(4,8,10,9)(11,13)(12,18)(14,25)(15,23)(16,24)(17,26)(19,22)(20,21);(11,14,17,20,16,19,15,12)(13,22,23,21,24,25,26,18);(3,6,7,5)(4,9,10,8)(11,15,16,17)(12,19,20,14)(13,23,24,26)(18,22,21,25)

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268165, #72268169, #72268189
maximal normal subgroup of#72268165, #72268169, #72268189

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352987);