#8664702

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	#38525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4364 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8571898);