#8990332

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

4372 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8897528);