#242753

order	512
nilpotency class	3
isoclinism	isoclinic class of #242745
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
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

706 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)(4,10)(5,8)(6,9)(7,13)(12,15)(14,16)(17,18,21,25,22,26,20,29)(19,23,27,31,28,32,30,24);(1,3,4,13,6,12,5,14)(2,7,8,11,9,16,10,15)(17,19)(18,23)(20,30)(21,27)(22,28)(24,29)(25,31)(26,32);(2,8)(3,12)(4,5)(7,11)(9,10)(15,16)(17,19)(18,24)(20,27)(21,30)(22,28)(23,29)(25,32)(26,31);(2,9)(3,12)(8,10)(13,14)(17,20,22,21)(18,25,26,29)(19,27,28,30)(23,24,32,31)

Polycyclic presentation:

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

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, 149949);