Finite Group Databasebeta

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Cayley diagram

C2×#9934 #234812

order512
nilpotency class3
isoclinismisoclinic class of #9930
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9934

Subgroups

subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2615C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72231648, C2×#234827
maximal normal subgroup of#72231648, C2×#234827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142008);