Finite Group Databasebeta

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Cayley diagram

C23×#440 #10574134

order512
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C214

Aliases

direct productC23×#440

Subgroups

subgroupG/H
center, socleC25C2×D4
commutatorC4C25×C4
Frattini, cosocleC2×C4C26
prime radicalC24×D4C22
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C25 ◁ C23×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

12706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC23×#3114, C23×C4×AGL(1,ℤ/8), C23×#3265, …
maximal normal subgroup ofC23×#3114, C23×C4×AGL(1,ℤ/8), C23×#3265, …
maximal quotient ofC23×#1762, C22×#21453, C22×#21528, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481330);