#8824147

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	#59819	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4292 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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, 8731343);