#238358

order	512
nilpotency class	3
isoclinism	isoclinic class of #238354
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
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

798 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)(4,8)(5,13)(6,9)(7,11)(12,16)(14,15)(17,18)(19,27,30,23)(20,24)(21,26)(22,25)(28,29,32,31);(4,5)(7,15)(8,13)(11,14)(17,18)(19,28,30,32)(20,24)(21,26)(22,25)(23,29,27,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(17,19)(18,23)(20,29)(21,31)(22,30)(24,28)(25,27)(26,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,16,15,10)(17,20,22,21)(18,24,25,26)(19,29,30,31)(23,28,27,32)

Polycyclic presentation:

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

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, 145554);