Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395248 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14722 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1715 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)(4,9)(5,18)(6,14)(8,11)(10,20)(12,22)(13,19)(15,23)(16,24)(17,25)(21,27)(26,31)(28,32)(29,30);(1,3,7,2,8,14,6,11)(4,9,22,24,17,25,16,12)(5,13,21,10,20,27,19,18)(15,23,31,32,29,30,28,26);(1,4)(2,9)(3,12)(5,15)(6,16)(7,22)(8,17)(10,23)(11,25)(13,26)(14,24)(18,30)(19,28)(20,29)(21,31)(27,32);(1,5,6,19,8,20,7,21)(2,10,3,13,11,18,14,27)(4,15,16,28,17,29,22,31)(9,23,12,26,25,30,24,32)Polycyclic presentation:
2^9|b^2=ei,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72219683, #72219690, … |
| maximal normal subgroup of | #72215931, #72219683, #72219690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302445);