#8324142

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

4508 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231338);