#285012

order	512
nilpotency class	3
isoclinism	isoclinic class of #284987
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

639 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)(3,10,12,16)(4,6,14,8)(7,13,15,11)(17,18,21,27,23,26,22,28)(19,24,29,25,20,31,30,32);(2,6)(4,14)(7,10)(8,9)(11,13)(15,16)(17,19,21,29,23,20,22,30)(18,24,27,25,26,31,28,32);(1,2)(3,7)(4,8)(5,9)(6,14)(10,11)(12,15)(13,16)(17,20,23,19)(18,25,26,32)(21,29,22,30)(24,28,31,27);(1,3,5,12)(2,7,9,15)(4,13,14,11)(6,10,8,16)(18,26)(24,31)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192208);