Finite Group Databasebeta

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Cayley diagram

C2×#24259 #6442634

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#24259

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#546C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5452 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);(4,5)(7,8)(9,10)(11,12)(13,21)(14,17)(15,20)(16,19)(18,24)(22,25)(23,26);(3,4,6,5)(12,17)(13,22)(15,16)(18,21)(23,24)(25,26);(8,10)(11,13,14,23)(12,18,17,25)(15,22,16,24)(19,26,20,21);(3,5,6,4)(12,17)(13,23)(19,20)(22,24)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6454101, C2×#6453973, C2×#6442688
maximal normal subgroup ofC2×#6454101, C2×#6453973, C2×#6442688

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349830);