#7972908

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

4332 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)(9,10,14,16)(11,17,12,15)(13,18,22,21)(19,24,20,23)(25,26,31,33)(27,32,30,34)(28,29,37,40)(35,36,38,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,12)(10,15,16,17)(13,20,22,19)(18,24,21,23)(25,27,31,30)(26,32,33,34)(28,35,37,38)(29,36,40,39);(1,3,5,4)(2,7,6,8)(10,16)(11,12)(18,21)(19,20)(26,33)(28,37)(32,34)(35,38);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,28)(26,29)(27,35)(30,38)(31,37)(32,36)(33,40)(34,39);(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,24)(20,23)(25,29)(26,28)(27,36)(30,39)(31,40)(32,35)(33,37)(34,38);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,30,31,27)(26,34,33,32)(28,38,37,35)(29,39,40,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=g,f^2=gh,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, 7880104);