#8339881

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	#35521	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4342 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,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,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8247077);