#6671338

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25189	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578534);