#8994055

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

4591 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8901251);