#8463224

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

4338 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370420);