#238924

order	512
nilpotency class	3
isoclinism	isoclinic class of #238921
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

872 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)(3,4,8,13)(5,11,16,10)(7,15,14,12)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,30)(29,32);(1,3)(2,7)(4,11)(5,15)(6,8)(9,14)(10,13)(12,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,30)(25,31)(26,32);(1,4,5,14,6,13,16,7)(2,8,10,15,9,3,11,12)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(2,9)(4,13)(7,14)(10,11)(17,20,22,21)(18,24,26,25)(19,28,29,27)(23,30,32,31)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,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=gi,d^c=dhi,f^c=fhi,g^c=ghi

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, 146120);