#8093211

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	#45785	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4488 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)(3,7)(4,6)(5,8)(10,14)(11,12)(18,21)(19,20)(26,30)(31,36)(32,37)(38,40);(1,2,5,8)(3,6,4,7)(9,10,15,14)(11,17,12,16)(13,21,22,18)(19,23,20,24)(27,33)(28,35)(31,36)(32,37);(9,11,15,12)(10,16,14,17)(13,19,22,20)(18,23,21,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,4)(2,6,8,7)(9,12,15,11)(10,17,14,16)(13,20,22,19)(18,24,21,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,2)(3,6)(4,7)(5,8)(9,13,15,22)(10,18,14,21)(11,19,12,20)(16,23,17,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40);(1,4,5,3)(2,7,8,6)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40)

Polycyclic presentation:

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

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, 8000407);