Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417620 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12)(4,16)(5,7)(8,21)(9,24)(10,18)(13,23)(15,19)(17,20)(22,28)(25,26)(27,30);(1,2)(3,13)(4,9)(5,18)(6,11)(7,10)(8,15)(12,21)(14,22)(16,24)(17,26)(19,23)(20,25)(27,32)(28,29)(30,31);(1,3)(2,8)(4,14)(5,19)(6,15)(7,12)(9,22)(10,23)(11,13)(16,30)(17,29)(18,21)(20,27)(24,31)(25,32)(26,28);(1,4)(2,9)(3,14)(5,16)(6,17)(7,20)(8,22)(10,25)(11,26)(12,27)(13,28)(15,29)(18,24)(19,30)(21,31)(23,32)Polycyclic presentation:
2^9|f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219485, #72219491, … |
| maximal normal subgroup of | #72213913, #72219485, #72219491, … |
| maximal quotient of | C2×#417620, #72251296, #72255732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324816);