Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38910 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18,23,26)(19,27,28,32)(20,24,30,29)(21,25,22,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(19,28)(20,30)(24,29)(27,32);(3,8)(5,7)(10,13)(11,15)(17,19,21,24,23,28,22,29)(18,20,31,32,26,30,25,27);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,25)(21,32)(22,27)(23,30)(26,29)(28,31);(10,13)(14,16)(17,21,23,22)(18,25,26,31)(19,24,28,29)(20,27,30,32)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308951, #72308996 |
| maximal normal subgroup of | #72308951, #72308996 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6847500);