#8075346

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

4476 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,e^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, 7982542);