#484989

order	512
nilpotency class	4
isoclinism	isoclinic class of #484989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60937	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

699 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 699 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,e^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=egi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392185);