Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1969 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,31)(23,27,33,36)(24,30,25,29)(28,34,35,32);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,27)(24,34)(25,32)(26,33)(28,29)(30,35)(31,36);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23,26,33)(22,28,31,35)(24,32,25,34)(27,30,36,29);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,23)(22,27)(24,34)(25,32)(26,33)(28,29)(30,35)(31,36);(6,12)(7,14)(11,20)(13,15)(21,24,26,25)(22,29,31,30)(23,32,33,34)(27,35,36,28)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72280867, #72280939, #72307467, … |
| maximal normal subgroup of | #72280867, #72280939, #72307467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6941908);