#238143

order	512
nilpotency class	3
isoclinism	isoclinic class of #233936
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145339);