Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #381483 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14299 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1399 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)(3,9,16,25)(4,10)(5,8)(6,12)(7,11)(13,21,15,24)(14,23,30,32)(17,31,29,26)(18,22)(19,28)(20,27);(2,8)(3,13)(5,7)(10,22)(11,12)(14,29)(15,16)(17,30)(18,20)(21,24)(26,31)(27,28);(1,3)(2,9)(4,17)(5,13)(6,16)(7,15)(8,21)(10,26)(11,24)(12,25)(14,18)(19,29)(20,30)(22,32)(23,27)(28,31);(1,4)(2,10)(3,14,16,30)(5,18)(6,19)(7,20)(8,22)(9,23,25,32)(11,27)(12,28)(13,29,15,17)(21,26,24,31)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216059, #72216077, #72216203, … |
| maximal normal subgroup of | #72216059, #72216077, #72216203, … |
| maximal quotient of | C2×#381483, #72252206, #72255649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288679);