Finite Group Databasebeta

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Cayley diagram

C2×#31483 #7504814

order512
nilpotency class3
isoclinismisoclinic class of #30412
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#31483

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

1862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6741717, C2×#6692620, C2×#7505487
maximal normal subgroup ofC2×#6741717, C2×#6692620, C2×#7505487

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7412010);