#8096669

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003865);