Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469685 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16660 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,15)(5,17)(7,11)(8,23)(10,18)(12,28)(14,19)(16,20)(21,31)(24,26)(27,32)(29,30);(1,2)(3,11)(4,8)(5,18)(6,9)(7,13)(10,17)(12,27)(14,22)(15,26)(16,24)(19,25)(20,23)(21,29)(28,32)(30,31);(1,3)(2,7,9,11)(4,12,16,29)(5,19)(6,13)(8,21)(10,25,18,22)(14,17)(15,30,20,28)(23,31)(24,27)(26,32);(1,4)(2,8)(3,12,13,29)(5,20)(6,16)(7,21,11,27)(9,24)(10,26)(14,30,19,28)(15,17)(18,23)(22,31,25,32)Polycyclic presentation:
2^9|c^2=f,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72339676, #72339680, #72339682 |
| maximal normal subgroup of | #72339676, #72339680, #72339682 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376881);