#6466882

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 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:

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

Polycyclic presentation:

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

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, 6374078);