#8092029

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

4387 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)(11,14)(12,17)(13,18)(15,16)(19,21)(20,24)(22,23)(26,31)(27,33)(32,39)(34,36);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,14,16,17)(13,19,22,20)(18,21,23,24)(25,26,30,31)(27,29,33,38)(28,32,35,39)(34,37,36,40);(1,4,5,2)(3,7,8,6)(10,16)(14,17)(18,23)(21,24)(28,35)(32,39)(34,36)(37,40);(9,12,15,11)(10,17,16,14)(13,19,22,20)(18,21,23,24)(25,27)(26,29)(28,36)(30,33)(31,38)(32,40)(34,35)(37,39);(9,13,15,22)(10,18,16,23)(11,19,12,20)(14,21,17,24)(25,28,30,35)(26,32,31,39)(27,34,33,36)(29,37,38,40);(9,14,15,17)(10,11,16,12)(13,21,22,24)(18,19,23,20)(25,29)(26,33)(27,31)(28,37)(30,38)(32,36)(34,39)(35,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 7999225);