#236327

order	512
nilpotency class	3
isoclinism	isoclinic class of #236309
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1227 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:

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

Polycyclic presentation:

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

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, 143523);