#484576

order	512
nilpotency class	4
isoclinism	isoclinic class of #484576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60771	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

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

normal subgroups only

This group has 839 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:

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

Polycyclic presentation:

2^9|a^2=gh,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^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, 391772);