#240946

order	512
nilpotency class	3
isoclinism	isoclinic class of #10549
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
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

867 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)(3,11)(4,5)(6,9)(7,10)(8,13)(12,15)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,31)(29,32);(1,3,7,15)(2,8,10,16)(4,11,9,12)(5,14,6,13)(17,18)(19,28)(20,24)(21,25)(22,26)(23,29)(27,31)(30,32);(1,4,5,2,7,9,6,10)(3,12,13,8,15,11,14,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,27)(25,28)(26,32);(17,20,22,21)(18,24,26,25)(19,29,31,30)(23,27,32,28)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=ghi,f^c=fhi,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, 148142);