#240813

order	512
nilpotency class	3
isoclinism	isoclinic class of #240803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60968	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 148009);