#9015385

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
prime radical	#42101	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922581);