#8601036

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8508232);