Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,15,11,16,10,12)(7,17,19,13,21,26,20,29)(8,22,23,14,25,27,24,30)(18,32,33,36,34,28,35,31);(1,2,3,4)(6,12)(9,10)(13,26)(14,27)(15,16)(17,29)(19,20)(22,30)(23,24)(28,36)(31,32)(33,35);(5,7)(6,13)(8,18)(9,20)(10,19)(11,21)(12,26)(14,28)(15,17)(16,29)(22,31)(23,35)(24,33)(25,34)(27,36)(30,32);(5,8)(6,14)(7,18)(9,24)(10,23)(11,25)(12,27)(13,28)(15,22)(16,30)(17,31)(19,35)(20,33)(21,34)(26,36)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72281002, #72281084, #72281232, … |
| maximal normal subgroup of | #72281002, #72281084, #72281232, … |
| maximal quotient of | #72281672 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299901);