#8824154

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	#38780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4224 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gi,e^2=i,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, 8731350);