#8463181

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370377);