Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,26,11,14,10,15)(7,16,18,28,20,31,19,12)(8,21,23,30,25,13,24,32)(17,33,22,27,35,29,34,36);(1,2)(3,4)(5,7)(6,12)(8,22)(9,18)(10,19)(11,20)(13,29)(14,28)(15,31)(16,26)(17,24)(21,33)(23,35)(25,34)(27,30)(32,36);(2,4)(5,8,11,25)(6,13,14,21)(7,17,20,35)(9,24,10,23)(12,27,28,36)(15,32,26,30)(16,33,31,29)(18,34,19,22);(6,14)(7,18)(9,10)(12,16)(13,21)(17,34)(19,20)(22,35)(23,24)(27,33)(28,31)(29,36)Polycyclic presentation:
2^9|a^2=ef,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72238736, #72264921, #72264949, … |
| maximal normal subgroup of | #72238736, #72264921, #72264949, … |
| maximal quotient of | #72238745 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326074);