#240776

order	512
nilpotency class	3
isoclinism	isoclinic class of #240771
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

933 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)(3,7)(4,6)(5,9)(8,12)(10,13)(11,14)(15,16)(18,22)(19,25)(20,28)(23,30)(24,29)(27,31);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,18,21,24)(19,26,27,32)(20,29,28,22)(23,31,30,25);(1,3,4,10,5,11,12,16)(2,7,8,15,9,14,6,13)(17,19,20,25,21,27,28,31)(18,23,22,26,24,30,29,32);(19,27)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|b^2=i,c^2=e,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,e^b=eh,f^b=fi,g^b=gh,d^c=di,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147972);