#7955046

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7862242);