Finite Group Databasebeta

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Cayley diagram

C2×#58197 #8104490

order512
nilpotency class2
isoclinismisoclinic class of #53390
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#58197

Subgroups

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

6390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72326959, #72326975, C2×#8968306, …
maximal normal subgroup of#72326959, #72326975, C2×#8968306, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8011686);