#8648975

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=ghi,e^2=h,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, 8556171);