Finite Group Databasebeta

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Cayley diagram

C2×#15113 #608293

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#15113

Subgroups

subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

7218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#4312858, #72250259, #72250260, …
maximal normal subgroup ofC2×#4312858, #72250259, #72250260, …
maximal quotient of#72250791, C2×#157223, C2×#157423

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515489);