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 | |
6110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(11,22)(23,33)(24,34)(35,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,23)(16,24)(17,22)(18,25)(19,26)(20,27)(21,28)(29,35)(30,33)(31,36)(32,34);(2,4)(5,7)(6,11)(8,15)(9,16)(10,17)(12,23)(13,24)(14,22)(18,29)(19,30)(20,31)(21,32)(25,35)(26,33)(27,36)(28,34);(2,4)(5,8,10,19)(6,12,14,26)(7,15,17,30)(9,20,21,18)(11,23,22,33)(13,27,28,25)(16,31,32,29)(24,36,34,35);(5,9,10,21)(6,13,14,28)(7,16,17,32)(8,18,19,20)(11,24,22,34)(12,25,26,27)(15,29,30,31)(23,35,33,36)Polycyclic presentation:
2^9|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, #72262403, #72282605, … |
| maximal normal subgroup of | #72256279, #72262403, #72282605, … |
| maximal quotient of | #72306403, #72306488, #72306502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344830);