#240086

order	512
nilpotency class	3
isoclinism	isoclinic class of #240081
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 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)(3,11)(4,13)(5,9)(6,15)(7,10)(8,12)(14,16)(17,18,21,24)(19,25,26,32)(20,28,30,23)(22,29,31,27);(1,3)(2,8)(4,9)(5,12)(6,16)(7,13)(10,14)(11,15);(1,4,6,8,7,11,5,14)(2,3,9,12,10,13,15,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(1,5,7,6)(2,9,10,15)(3,12,13,16)(4,14,11,8)(19,26)(22,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=di,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, 147282);