Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384259 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2333 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14417 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1458 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)(3,11,13,7)(4,15,18,8)(5,19)(6,9)(10,21)(12,27)(14,23,20,26)(16,24)(17,25,32,30)(22,28)(29,31);(1,3)(2,7)(4,16)(5,20)(6,13)(8,24)(9,11)(10,26)(12,17)(14,21)(15,29)(18,31)(19,23)(22,25)(27,30)(28,32);(1,4,5,17,6,18,21,32)(2,8,19,30,9,15,10,25)(3,12,14,16,13,28,20,31)(7,22,26,29,11,27,23,24);(7,11)(22,27)(23,26)(24,29)Polycyclic presentation:
2^9|a^2=h,c^2=eg,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216181, #72216320, … |
| maximal normal subgroup of | #72216029, #72216181, #72216320, … |
| maximal quotient of | #72251324, C2×#385872, #72256313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293068);