#8934881

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

4430 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8842077);