Finite Group Databasebeta

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Cayley diagram

C2×#31285 #7504290

order512
nilpotency class3
isoclinismisoclinic class of #31282
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#31285

Subgroups

subgroupG/H
center, socleC22#2684
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(3,4,10,7)(5,15,19,28)(6,20,24,30)(8,14,9,13)(11,18,25,17)(12,26,22,23)(16,29,21,27)(31,33,34,32);(3,5,9,18,10,19,8,17)(4,11,14,28,7,25,13,15)(6,21,23,33,24,16,26,32)(12,29,20,34,22,27,30,31);(3,6)(4,12)(5,16)(7,22)(8,23)(9,26)(10,24)(11,27)(13,20)(14,30)(15,31)(17,32)(18,33)(19,21)(25,29)(28,34);(3,7,10,4)(5,15,19,28)(6,22,24,12)(8,13,9,14)(11,18,25,17)(16,31,21,34)(20,26,30,23)(27,33,29,32)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72306450, #72306460, #72306464
maximal normal subgroup of#72306450, #72306460, #72306464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411486);