#8824422

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	#45385	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4301 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:

(2,4)(6,8)(9,10)(11,16)(12,18)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26,29,32)(27,33,35,30)(28,31,37,40)(34,39,36,38);(3,7)(6,8)(9,11,15,12)(10,16,17,18)(13,22,19,20)(14,23,24,21)(25,27,29,35)(26,30,32,33)(28,36,37,34)(31,39,40,38);(1,2,5,4)(3,6,7,8)(10,17)(11,12)(13,19)(21,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,37)(39,40);(9,12,15,11)(10,18,17,16)(13,22,19,20)(14,23,24,21)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13,15,19)(10,14,17,24)(11,20,12,22)(16,21,18,23)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39);(1,4,5,2)(3,8,7,6)(9,14)(10,19)(11,21)(12,23)(13,17)(15,24)(16,22)(18,20)(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=i,c^2=hi,d^2=gi,e^2=i,f^2=hi,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, 8731618);