#8211659

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	#59409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4316 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8118855);