Finite Group Databasebeta

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Cayley diagram

C2×#21638 #6373444

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#21638

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(4,5)(7,8)(9,12)(10,11)(13,14)(15,16,20,23,19,18,22,27)(17,21,25,28,24,26,29,30);(3,4,6,5)(8,11)(12,14)(15,17,19,24)(16,21,18,26)(20,25,22,29)(23,28,27,30);(3,5,6,4)(7,9,10,13)(8,12,11,14)(16,18)(17,24)(23,27)(25,29);(15,18,20,27,19,16,22,23)(17,21,25,28,24,26,29,30)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6373906, #72239393, #72264296, …
maximal normal subgroup ofC2×#6373906, #72239393, #72264296, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280640);