Finite Group Databasebeta

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Cayley diagram

C2×#9386 #200395

order512
nilpotency class3
isoclinismisoclinic class of #9385
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#9386

Subgroups

subgroupG/H
center, socleC23#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2543C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1744 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)(5,8)(9,10)(11,12)(13,16,20,25)(14,21,23,17)(15,18)(19,24)(22,26);(3,4,8,10,6,9,5,7)(11,13,15,20)(12,16)(14,22,23,19)(17,26)(18,25)(21,24);(4,9)(7,10)(11,14)(12,17)(13,19)(15,23)(16,24)(18,21)(20,22)(25,26)

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#200405, C2×#245567
maximal normal subgroup ofC2×#200405, C2×#245567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107591);