Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3176 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(22,27)(23,31)(25,32)(28,36)(29,34)(30,35);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,22,25,27,26,30,32,35)(23,28,31,33,29,36,34,24);(7,13)(10,17)(14,15)(18,19)(21,23,32,34,26,29,25,31)(22,28,35,24,30,36,27,33);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,24,26,33)(22,29,30,23)(25,28,32,36)(27,34,35,31);(6,12)(10,17)(11,20)(18,19)(23,29)(24,33)(28,36)(31,34)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250343, #72250360, #72256954, … |
| maximal normal subgroup of | #72250343, #72250360, #72256954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391909);