#8424233

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	#52529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,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, 8331429);