Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(15,22)(19,23)(24,27)(25,28);(5,7)(6,9)(8,12)(10,11)(13,14,18,21)(15,23,22,19)(16,17,20,26)(24,28,27,25);(5,7)(6,9)(8,12)(10,11)(13,15,18,22)(14,19,21,23)(16,24,20,27)(17,25,26,28);(5,7)(6,9)(8,12)(10,11);(5,7)(6,10)(8,12)(9,11)(13,16)(14,17)(15,24)(18,20)(19,25)(21,26)(22,27)(23,28);(5,6)(7,9)(8,11)(10,12)(13,17)(14,20)(15,25)(16,21)(18,26)(19,27)(22,28)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72304602, #72304604, #72304614, … |
| maximal normal subgroup of | #72304602, #72304604, #72304614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534420);