#8418672

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

4392 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)(3,7)(4,6)(5,8)(9,10,14,17)(11,13,16,18)(12,15,23,21)(19,22,20,24)(25,26)(27,31)(28,30)(29,33)(32,35)(34,38)(36,37)(39,40);(1,2,4,6)(3,5,8,7)(9,11)(10,13)(12,20)(14,16)(15,24)(17,18)(19,23)(21,22)(25,27,30,35)(26,31,28,32)(29,34,37,40)(33,38,36,39);(1,2)(3,5)(4,6)(7,8)(11,16)(13,18)(19,20)(22,24)(25,26,30,28)(27,32,35,31)(29,36,37,33)(34,38,40,39);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,21)(13,16)(14,17)(15,23)(19,22)(20,24)(25,28,30,26)(27,32,35,31)(29,36,37,33)(34,39,40,38);(9,12)(10,15)(11,19)(13,22)(14,23)(16,20)(17,21)(18,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,13,14,18)(10,16,17,11)(12,22,23,24)(15,20,21,19)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,f^2=gi,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, 8325868);