#8630875

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4510 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)(5,8)(6,7)(9,10,15,17)(11,13,14,16)(12,20,23,18)(19,24,21,22)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,3)(2,4)(5,6)(7,8)(9,11,15,14)(10,16,17,13)(12,21,23,19)(18,22,20,24)(26,27)(31,33)(32,34)(38,39);(1,4)(2,5)(3,7)(6,8)(10,17)(13,16)(18,20)(22,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3,6,5)(2,4,7,8)(9,12,15,23)(10,18,17,20)(11,19,14,21)(13,22,16,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,5,6,3)(2,4,7,8)(9,13,15,16)(10,11,17,14)(12,22,23,24)(18,19,20,21)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,11)(10,16,17,13)(12,19,23,21)(18,22,20,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=h,e^2=gi,f^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, 8538071);