#237012

order	512
nilpotency class	3
isoclinism	isoclinic class of #237007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144208);