#9683885

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9591081);