#471100

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

633 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=i,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=de,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378296);