#8688313

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	#2537	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

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, 8595509);