Finite Group Databasebeta

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Cayley diagram

C2×#21845 #6374376

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

Aliases

direct productC2×#21845

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,23,26,24,27,22,31)(21,28,30,25,29,34,32,33);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(21,29)(25,33)(28,34)(30,32);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21)(20,25)(22,32)(23,30)(24,29)(26,34)(27,33)(28,31);(4,10)(8,11)(9,18)(16,17)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#233571, #72303398, #72303447
maximal normal subgroup ofC2×#233571, #72303398, #72303447

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281572);