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Cayley diagram

C2×#53264 #8105681

order512
nilpotency class2
isoclinismisoclinic class of #53158
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#53264

Subgroups

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

6246 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);(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,17)(16,18)(21,25)(24,26);(4,7)(8,9)(12,13)(16,17)(20,23)(24,26);(11,12,15,13)(14,16,18,17)(19,20)(21,24)(22,23)(25,26);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(19,21)(20,24)(22,25)(23,26);(3,5,6,10)(4,8,7,9)(11,14,15,18)(12,16,13,17)(19,20)(21,24)(22,23)(25,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#8807309
maximal normal subgroup ofC2×#8807309

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012877);