#8069524

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

4361 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=h,e^2=h,f^2=gi,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, 7976720);