#8828707

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8735903);