Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437632 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
6222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,11,10)(7,12,18,17)(8,13,21,20)(9,14,22,23)(15,27,29,24)(16,25,30,31)(19,26,32,33)(28,35,36,34);(1,2)(3,4)(5,7)(6,12)(8,15)(9,16)(10,17)(11,18)(13,24)(14,25)(19,28)(20,27)(21,29)(22,30)(23,31)(26,34)(32,36)(33,35);(2,4)(5,8)(6,13)(7,15)(9,19)(10,20)(11,21)(12,24)(14,26)(16,28)(17,27)(18,29)(22,32)(23,33)(25,34)(30,36)(31,35);(2,4)(5,9,11,22)(6,14,10,23)(7,16,18,30)(8,19,21,32)(12,25,17,31)(13,26,20,33)(15,28,29,36)(24,34,27,35);(5,10,11,6)(7,17,18,12)(8,20,21,13)(9,14,22,23)(15,27,29,24)(16,25,30,31)(19,26,32,33)(28,34,36,35)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,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 | #72256279, #72262421, #72282641, … |
| maximal normal subgroup of | #72256279, #72262421, #72282641, … |
| maximal quotient of | #72306481, #72306487, #72306490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344833);