#241000

order	512
nilpotency class	3
isoclinism	isoclinic class of #240995
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

805 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148196);