Finite Group Databasebeta

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Cayley diagram

C2×#7949 #181141

order512
nilpotency class3
isoclinismisoclinic class of #1284
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#7949

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4,6,9,7,10,5,8)(11,12)(13,21,24,25)(14,18)(15,19)(16,20)(17,23,26,22);(4,8,10,9)(11,13)(12,17,20,26)(14,22)(15,23)(16,24)(18,25,19,21);(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,26,21)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239190, #72239268, #72239273
maximal normal subgroup of#72239190, #72239268, #72239273
maximal quotient of#72239228

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88337);