Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395244 |
| 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 | |
1720 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)(3,6)(4,13)(5,20)(7,11)(8,12)(9,18)(10,23)(14,22)(15,17)(16,29)(19,25)(21,26)(24,30)(27,32)(28,31);(1,3,7,12,8,11,6,2)(4,15,18,25,19,9,17,13)(5,14,22,20,23,26,21,10)(16,28,31,29,32,24,30,27);(1,4)(2,9)(3,13)(5,16)(6,17)(7,18)(8,19)(10,24)(11,25)(12,15)(14,27)(20,28)(21,30)(22,31)(23,32)(26,29);(1,5,6,21,8,23,7,22)(2,10,11,26,12,20,3,14)(4,16,17,30,19,32,18,31)(9,24,25,29,15,28,13,27)Polycyclic presentation:
2^9|b^2=ei,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72216141, #72219683, … |
| maximal normal subgroup of | #72215906, #72216141, #72219683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302441);