#6440851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439035
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 6348047);