#8701615

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	C₂×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4589 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=ghi,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, 8608811);