Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15)(7,16,14,10)(8,12,21,28)(9,22,24,13)(17,30,27,20)(18,25,29,23)(19,26,34,32)(31,33,36,35);(1,2)(3,4)(5,7,10,6,11,14,16,15)(8,17,20,12,21,27,30,28)(9,22,23,29,24,13,25,18)(19,32,33,36,34,26,35,31);(2,4)(5,8)(6,12)(7,17)(9,19)(10,20)(11,21)(13,26)(14,27)(15,28)(16,30)(18,31)(22,32)(23,33)(24,34)(25,35)(29,36);(5,9)(6,13)(7,18)(8,19)(10,25)(11,24)(12,26)(14,29)(15,22)(16,23)(17,31)(20,35)(21,34)(27,36)(28,32)(30,33)Polycyclic presentation:
2^9|a^2=i,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72251179, #72262403, #72262422, … |
| maximal normal subgroup of | #72251179, #72262403, #72262422, … |
| maximal quotient of | #72258158, #72258163, #72314510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325057);