#8932331

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

4436 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:

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

Polycyclic presentation:

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

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, 8839527);