#232856

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1250 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(1,3)(2,8)(4,12)(5,9)(6,14)(7,15)(10,13)(11,16)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(1,4)(2,9)(3,11)(5,16)(6,13)(7,8)(10,15)(12,14)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30);(1,5,7,6)(2,10,12,11)(3,9,15,14)(4,16,8,13)

Polycyclic presentation:

2^9|d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,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, 140052);