#8075429

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

4375 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=h,f^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, 7982625);