#8976315

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	#41028	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4419 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8883511);