#512006

order	512
nilpotency class	4
isoclinism	isoclinic class of #512006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32956
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3584	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,g^a=gi,h^a=hi,f^c=fhi,h^c=hi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419202);