#239929

order	512
nilpotency class	3
isoclinism	isoclinic class of #239921
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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,e^b=eh,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, 147125);