#8388794

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	#36410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4462 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,e^2=g,f^2=h,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, 8295990);