Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454670 |
| 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 | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,18)(3,12,14,8,15,25,19,22)(4,9,16,26,17,24,20,23)(13,28,31,32,30,27,29,21);(1,3)(2,8)(4,13)(5,19)(6,15)(7,14)(9,21)(10,25)(11,22)(12,18)(16,31)(17,30)(20,29)(23,27)(24,32)(26,28);(4,16)(5,7)(9,23)(10,18)(12,25)(13,29)(14,19)(17,20)(21,28)(24,26)(27,32)(30,31);(1,4)(2,9)(3,13)(5,20)(6,17)(7,16)(8,21)(10,26)(11,24)(12,27)(14,31)(15,30)(18,23)(19,29)(22,32)(25,28)Polycyclic presentation:
2^9|a^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237065, #72237069, #72237082, … |
| maximal normal subgroup of | #72237065, #72237069, #72237082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361866);