Finite Group Databasebeta

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Cayley diagram

C2×#60240 #10574292

order512
nilpotency class2
isoclinismisoclinic class of #3532
automorphism?
homologyAbelianization (H1) = C25×C4 · Schur multiplier (H2) = C213

Aliases

direct productC2×#60240

Subgroups

subgroupG/H
centerC22×C4C25
commutatorC22C25×C4
Frattini, cosocleC2×C4C26
socleC23C24×C4
prime radicalC2×D4×#41C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

7926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#10574299, C2×#10574489, C2×#10574513
maximal normal subgroup ofC2×#10574299, C2×#10574489, C2×#10574513

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10481488);