Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423431 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15314 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1338 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)(3,7)(4,8,16,25)(5,18)(6,10)(9,14)(11,20)(12,21,15,28)(13,22)(17,31,30,26)(19,23)(24,29,32,27);(1,3)(2,7)(4,15,16,12)(5,14)(6,13)(8,21,25,28)(9,18)(10,22)(11,23)(17,29,30,27)(19,20)(24,26,32,31);(1,4)(2,8,10,25)(3,12)(5,17)(6,16)(7,21,22,28)(9,24,23,32)(11,26,18,31)(13,15)(14,27)(19,29)(20,30);(1,3)(2,9)(4,12)(5,19)(6,13)(7,18)(8,24)(10,23)(11,22)(14,20)(15,16)(17,29)(21,31)(25,32)(26,28)(27,30)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237692, #72240435, #72240448, … |
| maximal normal subgroup of | #72237692, #72240435, #72240448, … |
| maximal quotient of | #72237698, #72237705 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330627);