#8107482

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	#34547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4466 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014678);