Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511861 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32899 |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #519 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,8)(6,20)(7,12)(9,18)(10,19)(13,27)(14,26)(16,24)(17,25)(21,23)(22,29)(28,31)(30,32);(2,8)(3,6)(5,19)(9,22)(10,12)(11,23)(13,17)(15,24)(16,29)(20,26)(25,31)(27,32);(1,3)(4,13)(5,14)(6,7)(8,10)(11,20)(16,28)(17,18)(19,21)(22,24)(25,27)(29,30);(1,4)(2,9,12,15)(3,13)(5,16)(6,17)(7,18)(8,22,10,24)(11,25,20,27)(14,28)(19,29)(21,30)(23,31,26,32)Polycyclic presentation:
2^9|d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72261089, #72261107, #72261130 |
| maximal normal subgroup of | #72261089, #72261107, #72261130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419057);