Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423435 |
| 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 | #23400 | 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)(3,8)(4,9,18,27)(5,12)(6,10)(7,11)(13,29,30,21)(14,24)(15,23)(16,22)(17,32,31,25)(19,28,20,26);(1,3)(2,8)(4,17,18,31)(5,15)(6,16)(7,14)(9,25,27,32)(10,22)(11,24)(12,23)(13,19,30,20)(21,28,29,26);(1,4)(2,9,11,27)(3,13,14,30)(5,19)(6,20)(7,18)(8,21)(10,26,12,28)(15,17,16,31)(22,32)(23,25)(24,29);(2,10)(5,6)(8,22)(9,26)(11,12)(15,16)(17,31)(19,20)(21,32)(23,24)(25,29)(27,28)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72216166, #72216217, … |
| maximal normal subgroup of | #72213514, #72216166, #72216217, … |
| maximal quotient of | #72250586, C2×#423435, #72265724 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330631);