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C24×#361 #72286503

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C215

Aliases

direct productC24×#361

Subgroups

subgroupG/H
center, socleC25#102
commutatorC8C25×C4
Frattini, cosocleC2×C8C26
prime radicalC25×D8C2
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C25 ◁ C25×C4 ◁ C25×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

50677 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);(10,14)(11,18)(12,19)(15,22)(16,20)(17,23)(21,24);(9,10,11,15,13,17,18,16)(12,20,19,23,21,22,24,14)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC24×AGL(1,ℤ/16), C23×#12956, C23×#12647, …
maximal normal subgroup ofC24×AGL(1,ℤ/16), C23×#12956, C23×#12647, …
maximal quotient ofC24×#1333, C23×#12736, C22×D4×#361, …

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), (10,14)(11,18)(12,19)(15,22)(16,20)(17,23)(21,24), (9,10,11,15,13,17,18,16)(12,20,19,23,21,22,24,14));