Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#577 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,19,11)(8,20,22,12)(9,14)(10,13)(16,26,18,24)(17,25)(21,33,34,27)(23,28)(29,32)(30,35)(31,36);(5,7)(6,11)(8,17)(9,16)(10,19)(12,25)(13,15)(14,24)(18,23)(20,29)(21,30)(22,32)(26,28)(27,35)(31,34)(33,36);(2,4)(7,16)(9,23)(11,24)(14,28)(15,26)(17,30)(18,19)(21,34)(25,35)(27,33)(29,36)(31,32);(5,8,10,22)(6,12,13,20)(7,17,19,32)(9,21,23,34)(11,25,15,29)(14,27,28,33)(16,30,18,31)(24,35,26,36)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313293, #72314237, #72314335, … |
| maximal normal subgroup of | #72313293, #72314237, #72314335, … |
| maximal quotient of | #72313296, #72313307, #72313337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266806);