#483027

order	512
nilpotency class	4
isoclinism	isoclinic class of #483025
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17675
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	#60714	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,f^2=g,g^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,f^c=fi,e^d=ei,f^d=fi,f^e=fi

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, 390223);