#8933097

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

4467 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8840293);