#8806757

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	#41027	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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)(3,4)(5,8)(6,7)(11,17)(14,18)(19,23)(21,24)(25,26,31,36)(27,35,30,32)(28,38,37,33)(29,34,39,40);(1,2,5,8)(3,6,7,4)(9,10,15,12)(11,14,17,18)(13,20,22,16)(19,24,23,21)(25,27,31,30)(26,32,36,35)(28,39,37,29)(33,40,38,34);(1,2,5,8)(3,4,7,6)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,28)(26,33)(27,29)(30,39)(31,37)(32,34)(35,40)(36,38);(1,3,5,7)(2,6,8,4)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,38)(39,40);(1,2)(3,6)(4,7)(5,8)(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29,31,39)(26,34,36,40)(27,37,30,28)(32,38,35,33);(1,4,5,6)(2,7,8,3)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30,31,27)(26,35,36,32)(28,39,37,29)(33,40,38,34)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=hi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8713953);