Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420407 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2004 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)(11,23)(15,29)(24,36)(30,31);(5,6)(7,15,17,29)(8,12)(9,14)(10,13)(11,18,23,34)(16,31,32,30)(19,26)(20,25)(21,28)(22,27)(24,33,36,35);(5,7)(6,11,13,23)(8,16)(9,18)(10,17)(12,24,25,36)(14,15,27,29)(19,33)(20,32)(21,35)(22,34)(26,31,28,30);(2,4)(5,8,10,20)(6,12,13,25)(7,16,17,32)(9,21,22,19)(11,24,23,36)(14,28,27,26)(15,30,29,31)(18,35,34,33)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72312562, #72312585, #72312596 |
| maximal normal subgroup of | #72312562, #72312585, #72312596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329987);