Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,15,18,11)(8,12)(9,13)(10,14)(16,29,32,23)(17,30,34,24)(19,25)(20,26)(21,27)(22,28)(31,36,33,35);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,23)(17,24)(19,25)(20,26)(21,27)(22,28)(29,32)(30,34)(31,35)(33,36);(2,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,23)(13,24)(14,15)(19,31)(20,32)(21,33)(22,34)(25,35)(26,29)(27,36)(28,30);(2,4)(5,8,10,20)(6,12,14,26)(7,16,18,32)(9,21,22,19)(11,23,15,29)(13,27,28,25)(17,33,34,31)(24,36,30,35);(5,9,10,22)(6,13,14,28)(7,17,18,34)(8,19,20,21)(11,24,15,30)(12,25,26,27)(16,31,32,33)(23,35,29,36)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280733, #72283147, #72283331, … |
| maximal normal subgroup of | #72280733, #72283147, #72283331, … |
| maximal quotient of | #72306395, #72306420, #72306430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345198);