#240661

order	512
nilpotency class	3
isoclinism	isoclinic class of #240653
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

931 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:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=h,f^2=i,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,d^c=dhi,f^c=fhi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147857);