#8509509

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	#55015	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416705);