#238054

order	512
nilpotency class	3
isoclinism	isoclinic class of #238050
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

885 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^c=fh,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, 145250);