#242679

order	512
nilpotency class	3
isoclinism	isoclinic class of #10603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#396
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 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)(4,9)(5,8)(6,10)(7,13)(12,15)(14,16)(17,18,21,25,22,26,20,24)(19,27,28,31,29,32,30,23);(1,3,4,13,6,12,5,14)(2,7,8,11,10,16,9,15);(2,8)(3,12)(4,5)(7,11)(9,10)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,31,32)

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,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149875);