#8132472

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	#59753	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8039668);