#239005

order	512
nilpotency class	3
isoclinism	isoclinic class of #239000
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	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

897 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)(3,13)(4,8)(5,11)(6,10)(7,12)(9,15)(14,16)(17,18)(19,27)(20,22)(21,25)(23,26)(24,29)(28,32)(30,31);(1,3)(2,8)(4,6)(5,15)(7,14)(9,12)(10,16)(11,13)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,27)(26,32);(1,4,5,14,7,15,6,3)(2,9,10,16,12,8,11,13)(17,20,21,26,23,25,22,18)(19,27,29,32,30,31,28,24);(17,21,23,22)(18,25,26,20)(19,28,30,29)(24,27,32,31)

Polycyclic presentation:

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

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, 146201);