#8159827

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	#59806	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4325 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8067023);