Finite Group Databasebeta

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Cayley diagram

C2×#21857 #6374422

order512
nilpotency class3
isoclinismisoclinic class of #3192
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#21857

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C22⋊C4
prime radicalC2×#3444C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3580 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);(4,7)(5,9)(11,12)(13,19,21,26)(14,18)(15,17)(16,20,24,22)(23,25);(3,4)(5,8)(6,7)(9,10);(4,7)(8,10)(11,13)(12,16)(14,22)(15,21)(17,24)(18,26)(19,25)(20,23);(3,5,6,9)(4,8,7,10)(12,17)(16,24)(18,25)(19,26)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303381, #72303398, C2×#233574
maximal normal subgroup of#72303381, #72303398, C2×#233574

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281618);