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C2×#497552 #72314201

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#497552

Subgroups

subgroupG/H
center, socleC23C22×#102
commutatorC2×C8C24×C4
Frattini, cosocleC22×C8C25
prime radicalC2×#61234C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ C22×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

5837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72265085, C2×#72264898, C2×#72265205
maximal normal subgroup ofC2×#72265085, C2×#72264898, C2×#72265205

GAP

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

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