#8807522

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	#38425	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8714718);