Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324468 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(17,18,22,25)(19,26,28,31)(20,24,21,30)(23,29,32,27);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(17,19,20,29,22,28,21,27)(18,23,30,31,25,32,24,26);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,32,26);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,28)(23,32)(26,31)(27,29)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285261 |
| maximal normal subgroup of | #72285261 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7243750);