#8800094

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

4349 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=hi,e^2=ghi,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, 8707290);