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C24×#365 #72286486

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C214

Aliases

direct productC24×#365

Subgroups

subgroupG/H
center, socleC25#106
commutatorC8C25×C4
Frattini, cosocleC2×C8C26
prime radicalC26SD16
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C25 ◁ C25×C4 ◁ C25×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

16917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC24×AGL(1,ℤ/16), C23×#12956, C23×#12929, …
maximal normal subgroup ofC24×AGL(1,ℤ/16), C23×#12956, C23×#12929, …
maximal quotient ofC24×#1368, C22×#483463, C23×#12929, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2), (3,4), (5,6), (7,8), (9,10,13,15,12,16,20,23,14,17,22,24,21,11,18,19), (9,11,12,17,14,16,21,10)(13,15,20,19,22,24,18,23));