Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481319 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1278 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)(5,6,9,14,10,15,24,26,11,16,8,21,23,30,22,12)(7,13,19,28,20,29,34,35,17,27,18,32,31,36,33,25);(6,12,16,26)(8,22)(9,24)(10,23)(13,25,27,35)(14,15,21,30)(18,33)(19,34)(20,31)(28,29,32,36);(1,2,3,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,17)(12,25)(14,28)(15,29)(16,27)(21,32)(22,33)(23,31)(24,34)(26,35)(30,36);(7,17)(13,27)(18,19)(20,31)(25,35)(28,32)(29,36)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239524, #72239538, #72239554, … |
| maximal normal subgroup of | #72239524, #72239538, #72239554, … |
| maximal quotient of | #72319106, #72319953, C2×#495259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402455);