#8724791

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4272 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)(3,7)(9,10)(11,19)(12,14)(13,16)(15,17)(18,23)(20,21)(22,24)(25,26,31,33)(27,32,36,30)(28,34,38,40)(29,39,37,35);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,17,19)(12,22,23,20)(14,21,18,24)(25,27)(26,32)(28,29)(30,33)(31,36)(34,39)(35,40)(37,38);(1,2)(3,6)(4,5)(7,8)(10,17)(12,23)(16,19)(20,22)(26,33)(27,36)(28,38)(35,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,18,17,14)(11,20,13,22)(16,21,19,24)(25,28)(26,34)(27,37)(29,36)(30,39)(31,38)(32,35)(33,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,19,17,16)(12,22,23,20)(14,21,18,24)(25,29,31,37)(26,35,33,39)(27,28,36,38)(30,34,32,40);(9,14,15,18)(10,12,17,23)(11,21,13,24)(16,22,19,20)(25,30,31,32)(26,27,33,36)(28,35,38,39)(29,40,37,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8631987);