#8649847

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

4379 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)(3,7)(4,8)(5,6)(25,26,32,34)(27,28,30,31)(29,33,38,40)(35,36,37,39);(2,6)(3,8)(9,10,14,17)(11,18,12,15)(13,16,21,24)(19,23,20,22)(25,27)(26,28)(29,37)(30,32)(31,34)(33,39)(35,38)(36,40);(1,3)(2,4)(5,8)(6,7)(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,24,23)(25,28,32,31)(26,27,34,30)(29,36,38,39)(33,35,40,37);(1,3)(2,4)(5,8)(6,7)(9,12,14,11)(10,15,17,18)(13,19,21,20)(16,23,24,22)(25,29)(26,33)(27,35)(28,36)(30,37)(31,39)(32,38)(34,40);(1,4)(2,3)(5,7)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,30,32,27)(26,28,34,31)(29,37,38,35)(33,36,40,39);(9,13,14,21)(10,16,17,24)(11,19,12,20)(15,22,18,23)(25,31,32,28)(26,27,34,30)(29,36,38,39)(33,37,40,35)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557043);