Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #62836 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3487 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,3,9)(4,10,16,27,18,26,14,23)(5,11,15,24,19,8,20,21)(17,25,28,32,30,22,29,31);(2,8)(5,19)(9,21)(10,22)(11,13)(12,24)(15,20)(17,30)(23,31)(25,26)(27,32)(28,29);(1,3,7,6)(2,9,13,12)(4,16,18,14)(5,15,19,20)(8,21,11,24)(10,23,26,27)(17,29,30,28)(22,31,25,32);(1,4)(2,10)(3,14)(5,17)(6,16)(7,18)(8,22)(9,23)(11,25)(12,27)(13,26)(15,28)(19,30)(20,29)(21,31)(24,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=dh,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72261130, #72261192 |
| maximal normal subgroup of | #72261130, #72261192 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400684);