Finite Group Databasebeta

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Cayley diagram

C2×#27713 #6457056

order512
nilpotency class3
isoclinismisoclinic class of #24269
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C26×C4

Aliases

direct productC2×#27713

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2190 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)(19,20,24,21,23,22,26,25);(4,7)(8,10)(11,13,14,17)(12,15,16,18)(19,21,24,22,23,25,26,20);(3,5,6,9)(4,8,7,10)(12,16)(15,18)(19,22)(20,23)(21,24)(25,26);(4,7)(8,10)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6692620, C2×#6460700, C2×#6720377
maximal normal subgroup ofC2×#6692620, C2×#6460700, C2×#6720377

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364252);