Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14893 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,12,20)(3,11,14,22,15,25,19,7)(4,16,17,31,18,9,23,24)(13,27,28,21,29,26,32,30);(1,3,12,14,6,15,5,19)(2,7,20,11,8,22,10,25)(4,13,23,28,18,29,17,32)(9,21,31,26,16,30,24,27);(2,8)(9,16)(10,20)(24,31);(1,4)(2,9,8,16)(3,13)(5,17)(6,18)(7,21,22,30)(10,24,20,31)(11,26,25,27)(12,23)(14,28)(15,29)(19,32)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285960, #72285996, #72286225 |
| maximal normal subgroup of | #72285960, #72285996, #72286225 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311729);