Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,16)(7,17,20,26)(8,21,24,14)(9,12,10,15)(13,25,29,18)(19,31,34,36)(22,27,23,30)(28,33,35,32);(1,2)(3,4)(6,12)(7,18)(8,22)(9,10)(14,21)(15,16)(17,26)(20,25)(23,24)(28,36)(31,35)(32,33);(5,7)(6,13)(8,19)(9,18)(10,25)(11,20)(12,26)(14,28)(15,17)(16,29)(21,35)(22,33)(23,32)(24,34)(27,36)(30,31);(2,4)(5,8)(6,14)(7,19)(9,23)(10,22)(11,24)(12,27)(13,28)(15,30)(16,21)(17,31)(18,32)(20,34)(25,33)(26,36)(29,35)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72255770, #72256284, #72256289 |
| maximal normal subgroup of | #72255770, #72256284, #72256289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324936);