#285398

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

584 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

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, 192594);