#8762033

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	#40413	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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)(3,5)(4,8)(6,7)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(2,7)(3,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(28,36)(31,38)(34,40)(37,39);(1,3,6,8)(2,4,7,5)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4,6,5)(2,3,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24);(1,2)(3,4)(5,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40);(1,5,6,4)(2,8,7,3)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=g,f^2=g,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, 8669229);