Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464074 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16500 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1469 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)(4,18)(5,13)(6,12)(7,11)(8,16)(9,17)(10,21)(15,24)(19,30)(20,29)(22,28)(23,25)(26,32)(27,31);(1,3)(2,8,12,25)(4,15)(5,16)(6,17)(7,23)(9,11,14,13)(10,24,29,27)(18,30,28,26)(19,20)(21,32)(22,31);(1,3)(2,9)(4,19)(5,23)(6,17)(7,16)(8,13)(10,27)(11,25)(12,14)(15,20)(18,30)(21,31)(22,32)(24,29)(26,28);(1,4)(2,10)(3,15)(5,21)(6,22)(7,20)(8,24)(9,26)(11,18)(12,29)(13,28)(14,30)(16,32)(17,31)(19,23)(25,27)Polycyclic presentation:
2^9|b^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72216037, #72216059, … |
| maximal normal subgroup of | #72213514, #72216037, #72216059, … |
| maximal quotient of | #72250589, C2×#464074, #72252203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371270);