#8700247

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4281 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607443);