#8934799

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
prime radical	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^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, 8841995);