Finite Group Databasebeta

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Cayley diagram

C2×#27811 #6457706

order512
nilpotency class3
isoclinismisoclinic class of #25443
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#27811

Subgroups

subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3058C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2092 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)(5,7)(6,8)(9,10)(11,12,15,19,16,20,14,17)(13,21,24,26,22,25,23,18);(3,5,6,9)(4,7,8,10)(12,17,20,19)(13,22)(18,21,26,25)(23,24);(4,8)(7,10)(11,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,26)(24,25);(11,13,14,23,16,22,15,24)(12,18,19,21,20,26,17,25)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6465969, #72226919, #72226925
maximal normal subgroup ofC2×#6465969, #72226919, #72226925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364902);