Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #24438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,20,25)(15,22,16,28)(17,23,33,42)(18,24,19,34)(21,30,37,27)(26,38,29,39)(31,44,43,36)(32,40,35,41);(5,6)(7,11)(8,10)(9,12)(13,15,20,16)(14,21,25,37)(17,31,33,43)(18,27,19,30)(22,24,28,34)(23,39,42,38)(26,35,29,32)(36,40,44,41);(2,4)(5,7,8,12)(6,9,10,11)(13,16,20,15)(14,22,25,28)(17,26,33,29)(18,27,19,30)(21,24,37,34)(23,39,42,38)(31,35,43,32)(36,41,44,40);(7,12)(9,11)(13,17,20,33)(14,23,25,42)(15,26,16,29)(18,32,19,35)(21,36,37,44)(22,38,28,39)(24,40,34,41)(27,43,30,31);(6,10)(9,11)(13,18,20,19)(14,24,25,34)(15,27,16,30)(17,32,33,35)(21,28,37,22)(23,40,42,41)(26,43,29,31)(36,39,44,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969946);