Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,22,26,30)(23,31,27,28)(24,34,25,35)(29,32,36,33);(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,23,26,27)(22,28,30,31)(24,32,25,33)(29,35,36,34);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(21,24)(22,29)(23,32)(25,26)(27,33)(28,35)(30,36)(31,34);(7,11)(10,12)(21,25,26,24)(22,29,30,36)(23,33,27,32)(28,35,31,34)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375263);