#8069489

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

4258 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 7976685);