Finite Group Databasebeta

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Cayley diagram

C2×#51324 #7706907

order512
nilpotency class2
isoclinismisoclinic class of #47826
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C213

Aliases

direct productC2×#51324

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

7454 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7614103);