Finite Group Databasebeta

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Cayley diagram

C2×#9950 #235036

order512
nilpotency class3
isoclinismisoclinic class of #9946
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC2×#9950

Subgroups

subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258163
maximal normal subgroup of#72258163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142232);