#7870503

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

4344 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)(9,10,14,17)(11,15,19,13)(12,20,22,16)(18,21,24,23)(25,26,32,37)(27,31,36,40)(28,35,30,33)(29,34,38,39);(1,2,5,6)(3,7,4,8)(25,27)(26,31)(28,29)(30,38)(32,36)(33,39)(34,35)(37,40);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,13,17,15)(12,18,22,24)(16,21,20,23)(25,28,32,30)(26,33,37,35)(27,38,36,29)(31,34,40,39);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,21)(22,24)(25,29,32,38)(26,34,37,39)(27,28,36,30)(31,35,40,33);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,17,20)(11,18,19,24)(13,21,15,23)(25,30,32,28)(26,35,37,33)(27,38,36,29)(31,34,40,39);(1,4,5,3)(2,8,6,7)(9,13,14,15)(10,11,17,19)(12,21,22,23)(16,18,20,24)(25,31,32,40)(26,36,37,27)(28,34,30,39)(29,35,38,33)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gi,e^2=gh,f^2=gh,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, 7777699);