Finite Group Databasebeta

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Cayley diagram

C2×#11373 #358577

order512
nilpotency class3
isoclinismisoclinic class of #1999
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#11373

Subgroups

subgroupG/H
center, socle, prime radicalC25C2×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C25 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

2610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#358652, C2×#359302
maximal normal subgroup ofC2×#358652, C2×#359302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265773);