#8592001

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	#40842	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8499197);