#8358726

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	#46133	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8265922);