#8755401

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

4181 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662597);