#8422206

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	#42679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 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)(11,14)(12,23)(13,15)(16,21)(17,20)(18,22)(19,24)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,2)(3,6)(4,7)(5,8)(9,11,16,20)(10,17,21,14)(12,15,22,19)(13,23,24,18)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,15)(13,17)(14,24)(16,22)(19,20)(21,23)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(1,3)(2,6)(4,8)(5,7)(9,13,16,24)(10,19,21,15)(11,18,20,23)(12,17,22,14)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,14,16,17)(10,20,21,11)(12,24,22,13)(15,18,19,23);(1,2)(3,5)(4,7)(6,8)(9,15,16,19)(10,13,21,24)(11,22,20,12)(14,18,17,23)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^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, 8329402);