#7806481

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	#34964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4399 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713677);