#240131

order	512
nilpotency class	3
isoclinism	isoclinic class of #240131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60805	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

916 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)(3,6)(4,12)(5,9)(7,11)(8,14)(10,15)(13,16)(17,18)(19,27)(20,22)(21,25)(23,26)(24,29)(28,31)(30,32);(1,3)(2,6)(4,11)(5,10)(7,12)(8,16)(9,15)(13,14)(17,19)(18,24)(20,31)(21,30)(22,29)(23,28)(25,27)(26,32);(1,3)(2,7)(4,13)(5,10)(6,12)(8,15)(9,16)(11,14)(17,20,21,26,23,25,22,18)(19,27,29,32,28,31,30,24);(3,10)(6,15)(7,16)(11,13)(19,28)(24,32)(27,31)(29,30)

Polycyclic presentation:

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

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, 147327);