Finite Group Databasebeta

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Cayley diagram

C2×#36567 #7630273

order512
nilpotency class2
isoclinismisoclinic class of D4×#120
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212

Aliases

direct productC2×#36567

Subgroups

subgroupG/H
center, socle, prime radicalC24C25
commutator, Frattini, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

5250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 512

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537469);