Finite Group Databasebeta

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Cayley diagram

C2×#11758 #481676

order512
nilpotency class4
isoclinismisoclinic class of #11757
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42

Aliases

direct productC2×#11758

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC2×#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

1336 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);(4,7,8,9)(11,12)(13,17)(14,21,23,16)(15,18)(19,26)(20,24,22,25);(3,4)(5,9)(6,8)(7,10)(12,16)(13,19)(17,24)(18,21)(20,22)(25,26);(11,13,15,19)(12,17,18,26)(14,22,23,20)(16,24,21,25)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72340416, #72340419, C2×#485614
maximal normal subgroup of#72340416, #72340419, C2×#485614

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388872);