Finite Group Databasebeta

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Cayley diagram

C22×#1995 #6372216

order512
nilpotency class3
isoclinismisoclinic class of #468
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC22×#1995

Subgroups

subgroupG/H
centerC23×C4C2×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC24C2×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#10015, #72279620, #72279655, …
maximal normal subgroup ofC22×#10015, #72279620, #72279655, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279412);