Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(10,14)(11,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(19,26)(22,29)(25,31)(28,32);(18,23)(22,29)(24,30)(28,32)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216158 |
| maximal normal subgroup of | #72216158 |
| maximal quotient of | #72216058 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367314);