#8534496

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	#38369	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4241 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=h,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441692);