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Cayley diagram

C2×#39838 #7637352

order512
nilpotency class2
isoclinismisoclinic class of #37644
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#39838

Subgroups

subgroupG/H
center, socleC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

6026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#7660918, C2×#7638259, C2×#7659424
maximal normal subgroup ofC2×#7660918, C2×#7638259, C2×#7659424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7544548);