#8460583

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,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, 8367779);