Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,21)(9,24)(10,14)(11,15)(12,17)(13,22)(18,27)(19,26)(20,28)(23,30)(25,29)(31,33)(32,35)(34,36);(5,7)(6,12)(8,18)(9,20)(10,17)(11,19)(13,27)(14,26)(15,28)(16,24)(21,31)(22,32)(23,34)(25,33)(29,36)(30,35);(7,17)(9,10)(12,26)(14,24)(16,28)(18,32)(19,20)(21,30)(22,25)(27,35)(31,36)(33,34);(1,3,4,2)(5,8)(6,13)(7,18)(9,25)(10,22)(11,23)(12,27)(14,30)(15,29)(16,31)(17,32)(19,34)(20,33)(21,24)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72312223, #72312227, #72312252 |
| maximal normal subgroup of | #72312223, #72312227, #72312252 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282955);