#242969

order	512
nilpotency class	3
isoclinism	isoclinic class of #242962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

663 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

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, 150165);