Finite Group Databasebeta

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Cayley diagram

C2×#8064 #181646

order512
nilpotency class3
isoclinismisoclinic class of #1504
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26

Aliases

direct productC2×#8064

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C2×#95
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

1746 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)(13,20)(14,18)(15,24)(16,19)(17,22)(21,26)(23,25);(4,8,10,9)(11,13,15,21,16,22,14,23)(12,17,18,26,19,20,24,25);(11,14,16,15)(12,18,19,24)(13,21,22,23)(17,25,20,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239498, #72239500, #72239507
maximal normal subgroup of#72239498, #72239500, #72239507
maximal quotient of#72239504

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88842);