Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417732 |
| 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×#3010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1946 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)(7,17)(8,22)(9,12)(10,15)(11,16)(13,18)(14,25)(19,31)(20,28)(21,26)(23,27)(24,30)(29,32)(33,36)(34,35);(6,12)(7,18)(8,23)(9,10)(13,28)(14,22)(15,16)(19,33)(20,21)(24,25)(29,36)(31,35);(2,4)(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,26)(14,29)(15,17)(16,28)(22,35)(23,34)(24,32)(25,33)(27,36)(30,31);(5,8,11,25)(6,14,16,22)(7,19,21,33)(9,24,10,23)(12,27,15,30)(13,29,28,35)(17,31,26,36)(18,32,20,34)Polycyclic presentation:
2^9|a^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72256515, #72256518, #72256521 |
| maximal normal subgroup of | #72256515, #72256518, #72256521 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326064);