#239919

order	512
nilpotency class	3
isoclinism	isoclinic class of #10440
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂×#3106	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

972 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(18,23)(19,26)(20,21)(24,31)(25,29)(27,28);(17,18)(19,24)(20,29)(21,23)(22,25)(26,30)(27,32)(28,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,20,27,22,28,21,26)(18,24,23,30,25,31,29,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)

Polycyclic presentation:

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

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, 147115);