#6499492

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35276	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406688);