#8936695

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	#40552	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8843891);