#8750630

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

4235 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657826);