#6444727

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2000 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgi,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6351923);