#6440848

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×#3008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2706 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)(11,20)(12,24)(13,28)(14,18)(15,19)(16,23)(17,27)(21,37)(22,33)(25,40)(26,36)(29,32)(30,35)(31,39)(34,38);(2,5)(6,8)(9,11,13,23,15,22,14,29)(10,16,18,20,19,32,28,33)(12,25,27,21,26,38,30,39)(17,34,24,37,35,40,36,31);(1,3,4,7)(2,6,5,8)(9,12,13,27,15,26,14,30)(10,17,18,24,19,35,28,36)(11,21,23,38,22,39,29,25)(16,31,20,34,32,37,33,40);(9,10)(11,20)(12,17)(13,18)(14,28)(15,19)(16,29)(21,34)(22,33)(23,32)(24,27)(25,31)(26,35)(30,36)(37,38)(39,40);(11,22)(12,26)(16,32)(17,35)(20,33)(23,29)(24,36)(27,30)

Polycyclic presentation:

2^9|b^2=g,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, 6348044);