#8417060

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

4199 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,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, 8324256);