#8898931

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

4678 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)(3,4)(5,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,26,29,31)(27,33,35,30)(28,32,37,39)(34,40,36,38);(2,6)(4,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,2,5,6)(3,7,8,4)(13,21)(16,18)(19,24)(22,23)(26,31)(30,33)(32,39)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,4,5,7)(2,3,6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,29,37)(26,32,31,39)(27,34,35,36)(30,38,33,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,26)(27,30)(28,32)(29,31)(33,35)(34,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806127);