#8722930

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	#59110	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4394 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:

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

Polycyclic presentation:

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

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, 8630126);