#8069673

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	#48693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4265 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976869);