#7883017

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

4559 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)(9,10,15,17)(11,20,23,16)(12,18,24,14)(13,21,22,19)(25,26)(27,36)(28,33)(29,39)(30,34)(31,35)(32,38)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,18)(14,22)(15,23)(17,20)(19,24)(25,27)(26,32)(28,30)(29,34)(31,38)(33,40)(35,36)(37,39);(10,17)(12,24)(16,20)(19,21)(25,28,31,39)(26,29,35,33)(27,37,38,30)(32,40,36,34);(1,2,5,4)(3,6,8,7)(9,12)(10,18)(11,21)(13,16)(14,17)(15,24)(19,23)(20,22)(25,28)(26,33)(27,30)(29,35)(31,39)(32,40)(34,36)(37,38);(9,13,15,22)(10,19,17,21)(11,18,23,14)(12,16,24,20)(25,29,31,33)(26,28,35,39)(27,34,38,40)(30,36,37,32);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,12,17,24)(11,22,23,13)(16,21,20,19)(25,30)(26,34)(27,28)(29,32)(31,37)(33,36)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=gh,e^2=h,f^2=gi,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, 7790213);