#8325711

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

4386 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=h,f^2=hi,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, 8232907);