Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45417 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,30,27,31,25,35)(23,29,33,36,24,32,34,28);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,24)(22,28,31,36)(25,33,26,34)(29,30,32,35);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,23)(22,29,31,32)(25,33,26,34)(28,30,36,35);(8,16)(11,20)(13,15)(17,19)(21,22,25,35,27,31,26,30)(23,32,34,36,24,29,33,28);(6,12)(7,14)(11,20)(13,15)(21,25,27,26)(22,30,31,35)(23,33,24,34)(28,32,36,29)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257078, #72257134, #72257193 |
| maximal normal subgroup of | #72257078, #72257134, #72257193 |
| maximal quotient of | #72308522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949794);