Finite Group Databasebeta

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Cayley diagram

C2×#8002 #181301

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

Aliases

direct productC2×#8002

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1726 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);(3,4,5,8,6,7,9,10)(11,12)(13,21)(14,18)(15,19)(16,20)(17,22)(23,26)(24,25);(4,7)(5,9)(11,13,15,23,16,24,14,22)(12,17,18,25,20,26,19,21);(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#72213342, #72213386, C2×#240064, …
maximal normal subgroup of#72213342, #72213386, C2×#240064, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88497);