Finite Group Databasebeta

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Cayley diagram

C2×#31266 #7504242

order512
nilpotency class3
isoclinismisoclinic class of #30396
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#31266

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3360 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:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6729222
maximal normal subgroup ofC2×#6729222

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411438);