#8158874

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

4450 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066070);