#8417302

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

4333 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,20)(13,18)(14,15)(17,21)(19,24)(22,23)(25,26)(27,28)(29,32)(30,37)(31,33)(34,39)(35,40)(36,38);(2,6)(3,7)(10,15)(12,21)(16,24)(18,22)(25,27,32,31)(26,33,29,28)(30,35,39,38)(34,40,37,36);(1,3,5,7)(2,4,6,8)(10,15)(13,23)(16,24)(17,20)(25,28)(26,31)(27,29)(30,36)(32,33)(34,38)(35,37)(39,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,22)(13,17)(14,19)(15,24)(18,21)(20,23)(25,29,32,26)(27,33,31,28)(30,37,39,34)(35,40,38,36);(9,12,14,21)(10,17,15,20)(11,18,19,22)(13,16,23,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40);(9,13,14,23)(10,18,15,22)(11,17,19,20)(12,16,21,24)(25,31,32,27)(26,33,29,28)(30,38,39,35)(34,40,37,36)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324498);