#8726224

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	#52292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4377 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=hi,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, 8633420);