Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418466 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16)(8,21)(9,13)(10,15)(11,19)(12,24)(14,25)(17,31)(18,27)(20,30)(22,36)(23,29)(26,34)(28,32)(33,35);(1,2,3,4)(5,7,9,19,10,20,25,18)(6,11,14,16,15,27,13,30)(8,22,23,33,24,17,32,34)(12,26,29,31,21,35,28,36);(5,8)(6,12)(7,17,20,22)(9,23)(10,24)(11,26,27,35)(13,28)(14,29)(15,21)(16,31,30,36)(18,33,19,34)(25,32);(2,4)(6,13)(7,18)(9,25)(11,27)(12,28)(14,15)(17,33)(19,20)(21,29)(22,34)(23,32)(26,35)Polycyclic presentation:
2^9|b^2=e,c^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72314443, #72314445, #72314468 |
| maximal normal subgroup of | #72314443, #72314445, #72314468 |
| maximal quotient of | #72314501 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330097);