#8618033

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

4636 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)(10,14)(15,17)(16,20)(22,23)(25,26,32,36)(27,29,30,33)(28,37,38,34)(31,35,40,39);(1,2,4,6)(3,5,8,7)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27)(26,33)(28,35)(29,36)(30,32)(31,37)(34,40)(38,39);(1,2)(3,5)(4,6)(7,8)(11,18)(15,17)(19,24)(22,23)(25,28,32,38)(26,34,36,37)(27,35,30,39)(29,31,33,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,29)(26,27)(28,31)(30,36)(32,33)(34,35)(37,39)(38,40);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,30,32,27)(26,29,36,33)(28,35,38,39)(31,34,40,37);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,31)(26,35)(27,37)(28,33)(29,38)(30,34)(32,40)(36,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8525229);