Finite Group Databasebeta

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Cayley diagram

C2×#27695 #6457046

order512
nilpotency class3
isoclinismisoclinic class of #24264
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#27695

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2548 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)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20,23,21,24,26,22,25);(4,7)(8,10)(11,13,14,17)(12,15,16,18)(19,21,22,20,24,25,23,26);(3,5,6,9)(4,8,7,10)(12,16)(15,18)(20,21)(22,23)(25,26);(4,7)(8,10)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6460952, C2×#6692620, C2×#6720228
maximal normal subgroup ofC2×#6460952, C2×#6692620, C2×#6720228

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364242);