#8075535

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

4472 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)(11,14)(12,17)(13,18)(15,16)(19,21)(20,24)(22,23)(25,26)(27,28)(29,33)(30,32)(31,34)(35,36)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(9,11,15,12)(10,14,16,17)(13,19,22,20)(18,21,23,24)(25,27)(26,29)(28,30)(31,35)(32,33)(34,37)(36,38)(39,40);(10,16)(14,17)(18,23)(21,24)(25,28,32,29)(26,33,30,27)(31,37,39,36)(34,35,38,40);(9,12,15,11)(10,17,16,14)(13,19,22,20)(18,21,23,24)(25,29,32,28)(26,33,30,27)(31,36,39,37)(34,35,38,40);(1,3,5,4)(2,7,6,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30,32,26)(27,28,33,29)(31,38,39,34)(35,36,40,37);(1,4,5,3)(2,8,6,7)(9,14,15,17)(10,11,16,12)(13,21,22,24)(18,19,23,20)(25,31)(26,34)(27,35)(28,36)(29,37)(30,38)(32,39)(33,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=h,e^2=g,f^2=gh,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, 7982731);