#8757119

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

4305 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)(4,5)(6,8)(9,10,15,17)(11,16,21,24)(12,18,20,14)(13,23,22,19)(27,33)(30,34)(35,39)(38,40);(2,5)(6,7)(9,11)(10,16)(12,22)(13,20)(14,23)(15,21)(17,24)(18,19)(25,26)(27,34)(28,36)(29,32)(30,33)(31,37)(35,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,21)(12,20)(19,23)(25,26,29,32)(27,30,33,34)(28,31,37,36)(35,38,39,40);(1,3)(2,6)(4,8)(5,7)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,19,24,23)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(9,13,15,22)(10,19,17,23)(11,20,21,12)(14,16,18,24)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(9,14,15,18)(10,20,17,12)(11,19,21,23)(13,16,22,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664315);