#105891

order	512
nilpotency class	3
isoclinism	isoclinic class of #105878
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,f^b=fh,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13087);