Finite Group Databasebeta

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C2×#135908 #72285271

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C27

Aliases

direct productC2×#135908

Subgroups

subgroupG/H
center, socleC24#442
commutatorC2×C8C23×C8
Frattini, cosocleC2×C4×C8C24
prime radicalC23×D8C2×C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ C22×#322 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

2057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 4 generators on 30 points:

(4,6)(7,8,9,11,10,12,13,14)(15,16,18,23,20,26,28,25)(17,21,24,30,22,29,27,19);(15,17)(16,21)(18,24)(19,25)(20,22)(23,30)(26,29)(27,28);(3,4,5,6)(16,22)(17,26)(18,28)(19,29)(21,30)(23,24)(25,27);(1,2)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72225051, C2×#72227052, #72286538, …
maximal normal subgroup ofC2×#72225051, C2×#72227052, #72286538, …
maximal quotient ofC2×#72225047, #72286538, C2×#72225056, …

GAP

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

gap> G := Group((4,6)(7,8,9,11,10,12,13,14)(15,16,18,23,20,26,28,25)(17,21,24,30,22,29,27,19), (15,17)(16,21)(18,24)(19,25)(20,22)(23,30)(26,29)(27,28), (3,4,5,6)(16,22)(17,26)(18,28)(19,29)(21,30)(23,24)(25,27), (1,2));