#8291661

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

4298 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=i,e^2=g,f^2=g,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, 8198857);