Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1226 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)(3,7,13,21)(4,8)(5,11)(6,10)(9,14,22,17)(12,27,28,19)(15,24)(16,25)(18,20)(23,29,26,31)(30,32);(1,3)(2,7,10,21)(4,12,15,28)(5,17)(6,13)(8,19)(9,20,22,11)(14,18)(16,31,30,29)(23,32)(24,27)(25,26);(1,4)(2,8)(3,12,13,28)(5,16)(6,15)(7,19,21,27)(9,26,22,23)(10,24)(11,25)(14,29,17,31)(18,30)(20,32);(1,3)(2,9,10,22)(4,12,15,28)(5,14)(6,13)(7,20,21,11)(8,23)(16,29,30,31)(17,18)(19,32)(24,26)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338620, #72338627, #72338636 |
| maximal normal subgroup of | #72338620, #72338627, #72338636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367316);