#8612273

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	#36426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4379 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8519469);