#8534893

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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)(3,7)(4,5)(6,8)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(2,5)(6,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,21)(22,24)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,13,17,18)(12,20,22,19)(15,23,24,21)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,23)(22,24)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(9,12,14,22)(10,15,17,24)(11,19,16,20)(13,21,18,23)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39);(9,13)(10,16)(11,17)(12,21)(14,18)(15,20)(19,24)(22,23)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=ghi,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, 8442089);