Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16255 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,10,5,12)(3,13,15,8,16,24,21,23)(4,9,18,26,20,28,19,27)(14,29,17,22,31,25,30,32);(1,3)(2,8)(4,17)(5,21)(6,16)(7,15)(9,25)(10,23)(11,24)(12,13)(14,19)(18,31)(20,30)(22,27)(26,32)(28,29);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,22)(10,28)(11,27)(12,26)(13,25)(15,30)(16,31)(17,21)(23,32)(24,29);(2,10)(4,18)(5,7)(8,23)(9,26)(14,30)(15,21)(17,31)(19,20)(22,29)(25,32)(27,28)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72216047, #72216154, #72216203, … |
| maximal normal subgroup of | #72216047, #72216154, #72216203, … |
| maximal quotient of | C2×#454668, #72255598, #72255664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361864);