#8107572

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014768);