#7972521

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

4547 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7879717);