#8635861

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	#42989	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=gh,f^2=gi,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, 8543057);