#8424268

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) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

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, 8331464);