Finite Group Databasebeta

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Cayley diagram

C23×#443 #10574133

order512
nilpotency class3
isoclinismisoclinic class of D8
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C214

Aliases

direct productC23×#443

Subgroups

subgroupG/H
centerC23×C8D4
commutatorC4C25×C4
Frattini, cosocleC2×C4C26
socleC24C4×D4
prime radicalC23×#135C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C8 ◁ C24×C8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

10690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC23×#2179, C23×#2171, C23×#1336
maximal normal subgroup ofC23×#2179, C23×#2171, C23×#1336

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481329);