#239290

order	512
nilpotency class	3
isoclinism	isoclinic class of #238025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146486);