Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,8)(9,11)(10,12)(13,14,19,24)(15,20,27,36)(16,28,31,38)(17,23,18,22)(21,29,37,30)(25,39,41,43)(26,34,32,35)(33,42,44,40);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,32)(18,26)(19,27)(21,33)(22,34)(23,35)(24,36)(28,39)(29,42)(30,40)(31,41)(37,44)(38,43);(2,4)(13,16,19,31)(14,21,24,37)(15,25,27,41)(17,30,18,29)(20,33,36,44)(22,28,23,38)(26,42,32,40)(34,39,35,43);(6,11)(7,12)(14,22)(17,18)(20,34)(21,28)(23,24)(26,32)(29,30)(33,39)(35,36)(37,38)(40,42)(43,44);(5,8,9,10)(6,7,11,12)(13,17,19,18)(14,23,24,22)(15,26,27,32)(16,29,31,30)(20,34,36,35)(21,28,37,38)(25,40,41,42)(33,43,44,39)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410008);