#7870472

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4259 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7777668);