Finite Group Databasebeta

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Cayley diagram

C2×#8128 #181414

order512
nilpotency class3
isoclinismisoclinic class of #1304
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#8128

Subgroups

subgroupG/H
centerC22×C4#95
commutatorC23C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C42
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1618 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);(4,7)(5,9)(11,12,15,20,16,19,14,18)(13,21,22,26,23,17,24,25);(3,4,5,8,6,7,9,10)(11,13)(12,17)(14,22)(15,24)(16,23)(18,25)(19,21)(20,26);(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239971, C2×#181415, #72240026
maximal normal subgroup of#72239971, C2×#181415, #72240026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88610);