#8285379

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	#35692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4648 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8192575);