Finite Group Databasebeta

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Cayley diagram

C22×#1574 #6344329

order512
nilpotency class3
isoclinismisoclinic class of #447
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC22×#1574

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC24C2×C4⋊C4
prime radicalC25C4⋊C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

2102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2);(3,4);(5,6,10,12)(7,8,11,9)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(5,7,9,6,10,11,8,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(5,8,10,9)(6,11,12,7)(15,21)(18,24)(22,27)(25,28)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#17834, C22×#17917, C22×#18022
maximal normal subgroup ofC22×#17834, C22×#17917, C22×#18022

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251525);