#7971651

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

4372 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)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(26,31)(28,37)(32,33)(34,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,26,30,31)(27,33,36,32)(28,29,37,39)(34,38,40,35);(1,3,4,7)(2,6,5,8)(10,12)(16,18)(17,20)(23,24)(25,27,30,36)(26,32,31,33)(28,34,37,40)(29,35,39,38);(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28)(26,29)(27,34)(30,37)(31,39)(32,35)(33,38)(36,40);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,27)(26,33)(28,34)(29,38)(30,36)(31,32)(35,39)(37,40);(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,29)(26,28)(27,35)(30,39)(31,37)(32,34)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=h,e^2=h,f^2=h,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, 7878847);