#108027

order	512
nilpotency class	3
isoclinism	isoclinic class of #108026
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1048 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)(3,4,8,6)(5,7)(9,10)(11,19)(12,23,24,33)(13,17)(14,18)(15,21)(16,25,31,27)(20,36,38,40)(22,32)(26,30)(28,34)(29,39,35,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15,18,19)(12,20)(13,26)(14,21)(16,29,31,35)(17,32,34,30)(22,28)(23,36,33,40)(24,38)(25,39)(27,37);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,33)(18,31)(19,35)(21,38)(22,39)(23,34)(25,28)(26,37)(30,36)(32,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,d^c=dhi,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, 15223);