Finite Group Databasebeta

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Cayley diagram

C2×#17608 #6343431

order512
nilpotency class3
isoclinismisoclinic class of #17603
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC2×#17608

Subgroups

subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3489C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

4478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6343491, #72261762
maximal normal subgroup ofC2×#6343491, #72261762

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250627);