#484983

order	512
nilpotency class	4
isoclinism	isoclinic class of #484983
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

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

normal subgroups only

This group has 695 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,e^2=g,g^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gi,h^a=hi,e^b=egi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 392179);