Finite Group Databasebeta

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Cayley diagram

C2×#62999 #10586044

order512
nilpotency class2
isoclinismisoclinic class of D4×H(5,𝔽2)
automorphism?
homologyAbelianization (H1) = C27 · Schur multiplier (H2) = C219

Aliases

direct productC2×#62999

Subgroups

subgroupG/H
center, socleC23C26
commutator, Frattini, cosocleC22C27
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C22 ▷ C1

34588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493240);