#8399474

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	#34977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4509 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:

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

Polycyclic presentation:

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

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, 8306670);