#9057921

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

4799 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965117);