Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456656 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16284 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1478 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)(3,7,12,22)(4,8)(5,17)(6,9)(10,20)(11,29,30,21)(13,27,18,23)(14,24,31,32)(15,28)(16,25)(19,26);(1,3)(2,7,9,22)(4,14)(5,18)(6,12)(8,24,25,32)(10,27,17,23)(11,15)(13,20)(16,31)(19,30)(21,28,29,26);(1,4)(2,8)(3,11)(5,19)(6,16)(7,21)(9,25)(10,28)(12,30)(13,31)(14,18)(15,20)(17,26)(22,29)(23,32)(24,27);(5,20)(8,25)(10,17)(11,30)(13,18)(15,19)(23,27)(24,32)Polycyclic presentation:
2^9|a^2=f,b^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285398, #72286231, #72286246, … |
| maximal normal subgroup of | #72285398, #72286231, #72286246, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363852);