#8422054

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8329250);