#284871

order	512
nilpotency class	3
isoclinism	isoclinic class of #284859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×AGL(1,ℤ/8)	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192067);