#8095867

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
prime radical	#45486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4323 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=h,f^2=h,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, 8003063);