#8550970

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

4453 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8458166);