#9863155

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4525 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,13)(14,15)(16,20)(18,23)(19,24)(21,22)(25,26,30,32)(27,31,35,39)(28,33,38,37)(29,36,40,34);(2,6)(3,7)(10,15)(12,21)(16,17)(18,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(1,3)(2,4)(5,7)(6,8)(26,32)(27,35)(33,37)(34,36);(1,4)(2,3)(5,8)(6,7)(9,11,14,20)(10,16,15,17)(12,18,21,24)(13,19,22,23)(25,27)(26,31)(28,36)(29,37)(30,35)(32,39)(33,40)(34,38);(9,12,14,21)(10,13,15,22)(11,18,20,24)(16,19,17,23)(25,28)(26,33)(27,36)(29,39)(30,38)(31,40)(32,37)(34,35);(9,13)(10,12)(11,19)(14,22)(15,21)(16,18)(17,24)(20,23)(25,29,30,40)(26,34,32,36)(27,37,35,33)(28,31,38,39)

Polycyclic presentation:

2^9|a^2=i,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770351);