#8096326

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	#45995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4306 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)(11,19)(12,14)(13,16)(15,18)(17,23)(20,24)(21,22)(25,26,32,34)(27,28,31,30)(29,37,40,33)(35,39,38,36);(1,2)(3,6)(4,7)(5,8)(9,11,15,13)(10,16,18,19)(12,20,23,22)(14,21,17,24)(25,27,32,31)(26,30,34,28)(29,38,40,35)(33,36,37,39);(1,3)(2,5)(4,8)(6,7)(25,28,32,30)(26,27,34,31)(29,36,40,39)(33,35,37,38);(9,12)(10,17)(11,20)(13,22)(14,18)(15,23)(16,24)(19,21)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,40)(34,37);(1,3)(2,6)(4,8)(5,7)(9,13,15,11)(10,16,18,19)(12,22,23,20)(14,21,17,24)(25,30,32,28)(26,27,34,31)(29,39,40,36)(33,35,37,38);(1,2)(3,5)(4,7)(6,8)(9,14,15,17)(10,12,18,23)(11,21,13,24)(16,20,19,22)(25,31,32,27)(26,30,34,28)(29,38,40,35)(33,39,37,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003522);