#8947705

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

4456 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8854901);