#8496831

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

4676 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)(3,7)(4,6)(5,8)(10,11)(12,19)(16,18)(20,24)(25,26,31,30)(27,32,37,36)(28,38,29,33)(34,40,35,39);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,17,19,15)(13,16,21,18)(20,23,24,22)(25,27,31,37)(26,32,30,36)(28,35,29,34)(33,40,38,39);(1,2)(3,5)(4,6)(7,8)(25,28,31,29)(26,33,30,38)(27,34,37,35)(32,39,36,40);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,29,31,28)(26,33,30,38)(27,35,37,34)(32,39,36,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,32)(28,34)(29,35)(30,36)(31,37)(33,39)(38,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,26)(27,36,37,32)(28,38,29,33)(34,40,35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,f^2=gi,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, 8404027);