Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394376 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2617 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)(3,14)(4,9)(5,12)(6,11)(7,13)(8,17)(10,20)(15,30)(16,23)(18,26)(19,25)(21,29)(22,28)(24,32)(27,31);(1,3,5,17,7,18,6,19)(2,8,11,14,13,25,12,26)(4,16,20,31,21,32,22,15)(9,27,28,23,29,30,10,24);(1,4)(2,9)(3,15)(5,22)(6,20)(7,21)(8,23)(10,11)(12,28)(13,29)(14,27)(16,19)(17,32)(18,31)(24,25)(26,30);(1,4)(2,10)(3,16)(5,20)(6,22)(7,21)(8,24)(9,11)(12,29)(13,28)(14,27)(15,19)(17,31)(18,32)(23,25)(26,30)Polycyclic presentation:
2^9|b^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216047, #72216112, … |
| maximal normal subgroup of | #72216037, #72216047, #72216112, … |
| maximal quotient of | #72237503, #72237516, C2×#394376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301572);