#8195966

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
prime radical	#59303	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103162);