Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,24,27,30,25,28)(23,31,34,33,32,29,36,35);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,28)(23,32)(24,30)(25,26)(29,35)(31,33);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23)(22,29)(24,35)(25,36)(26,34)(27,32)(28,33)(30,31);(8,16)(11,20)(13,15)(17,19)(21,24,25,22,27,28,26,30)(23,33,36,31,32,35,34,29);(6,12)(7,14)(11,20)(13,15)(21,25,27,26)(22,24,30,28)(23,34,32,36)(29,33,31,35)Polycyclic presentation:
2^9|a^2=fhi,c^2=gi,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280254, #72280346, #72281990, … |
| maximal normal subgroup of | #72280254, #72280346, #72281990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949422);