#8599189

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=h,e^2=g,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, 8506385);