#8088059

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=i,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=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, 7995255);