Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,24,30)(7,12,19,26,18,9,14,23)(8,13,22,29,21,28,34,36)(17,20,32,33,31,25,27,35);(1,2)(3,4)(5,7)(6,12)(8,17)(9,16)(10,19)(11,18)(13,25)(14,24)(15,26)(20,28)(21,31)(22,32)(23,30)(27,34)(29,35)(33,36);(2,4)(5,8)(6,13)(7,17)(9,20)(10,22)(11,21)(12,25)(14,27)(15,29)(16,28)(18,31)(19,32)(23,33)(24,34)(26,35)(30,36);(5,9,10,23,11,12,24,26)(6,14,15,7,16,19,30,18)(8,20,22,33,21,25,34,35)(13,27,29,17,28,32,36,31)Polycyclic presentation:
2^9|a^2=h,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249969, #72250492, #72252634, … |
| maximal normal subgroup of | #72249969, #72250492, #72252634, … |
| maximal quotient of | #72314526, C2×#375869, #72343019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283065);