#479803

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386999);