Finite Group Databasebeta

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Cayley diagram

C2×#38314 #7635574

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#38314

Subgroups

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

5962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542770);