#8505607

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

4723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=i,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412803);