Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,21,23,22,25,20,26)(19,27,29,24,30,32,28,31);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(17,19,22,30)(18,24,25,31)(20,29,21,28)(23,27,26,32);(18,25)(23,26)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249801, #72249804, #72249808, … |
| maximal normal subgroup of | #72249801, #72249804, #72249808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106157);