#8806927

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714123);