#8799477

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

4345 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8706673);