#8926666

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

4384 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)(3,7)(4,6)(5,8)(11,12)(13,21)(15,18)(16,23)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,2,4,6)(3,5,8,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,15,17,18)(13,20,21,19)(16,24,23,22)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,15,17,18)(13,20,21,19)(16,22,23,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,10)(11,15)(12,18)(13,16)(14,17)(19,22)(20,24)(21,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(9,13,14,21)(10,16,17,23)(11,19,12,20)(15,22,18,24)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833862);