Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423218 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15313 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1545 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)(3,13)(4,18)(5,11)(6,10)(7,12)(8,16)(9,21)(14,23)(15,25)(17,24)(19,26)(20,28)(22,27)(29,30)(31,32);(1,3,5,16,7,17,6,15)(2,8,10,13,12,25,11,24)(4,19,20,14,22,30,21,31)(9,26,27,32,28,29,18,23);(1,4)(2,9)(3,14,17,31)(5,21)(6,20)(7,22)(8,23,25,32)(10,18)(11,27)(12,28)(13,29,24,26)(15,30,16,19);(3,15)(5,6)(8,24)(10,11)(13,25)(14,30)(16,17)(18,27)(19,31)(20,21)(23,26)(29,32)Polycyclic presentation:
2^9|b^2=e,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72216192, #72216942, … |
| maximal normal subgroup of | #72215912, #72216192, #72216942, … |
| maximal quotient of | #72237173, C2×#423218, #72255697 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330414);