#8069532

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4210 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)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,26,32,31)(27,33,30,36)(28,35,29,34)(37,40,38,39);(1,2)(3,6)(4,5)(7,8)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(10,14)(15,21)(16,22)(23,24)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,29,32,28)(26,35,31,34)(27,37,30,38)(33,39,36,40);(1,3,4,7)(2,6,5,8)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,30,32,27)(26,36,31,33)(28,37,29,38)(34,39,35,40);(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,31,32,26)(27,36,30,33)(28,34,29,35)(37,39,38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=gh,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, 7976728);