#232681

order	512
nilpotency class	3
isoclinism	isoclinic class of #10026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1260 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139877);