#7973913

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

4221 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7881109);