Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16)(7,17,21,25)(8,13,23,29)(9,15,10,14)(12,18,27,20)(19,33,34,31)(22,30,24,28)(26,35,36,32);(1,2)(3,4)(7,18)(9,10)(12,25)(14,15)(17,27)(19,32)(20,21)(22,24)(26,33)(28,30)(31,36)(34,35);(1,3,4,2)(5,7)(6,12)(8,19)(9,18)(10,20)(11,21)(13,26)(14,25)(15,17)(16,27)(22,35)(23,34)(24,32)(28,31)(29,36)(30,33);(5,8)(6,13)(7,19)(9,24)(10,22)(11,23)(12,26)(14,30)(15,28)(16,29)(17,31)(18,32)(20,35)(21,34)(25,33)(27,36)Polycyclic presentation:
2^9|a^2=i,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317994, #72318531, #72318566 |
| maximal normal subgroup of | #72317994, #72318531, #72318566 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301737);