#239012

order	512
nilpotency class	3
isoclinism	isoclinic class of #239009
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 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)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(17,18,22,26)(19,23,29,32)(20,24,21,25)(27,30,28,31);(1,3,6,13)(2,7,9,8)(4,14,12,5)(10,16,15,11)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,30)(26,32);(1,4,5,3,6,12,14,13)(2,8,10,16,9,7,15,11)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(2,9)(7,8)(10,15)(11,16)

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,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,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, 146208);