#483309

order	512
nilpotency class	4
isoclinism	isoclinic class of #11994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1558
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1558
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

659 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 659 conjugacy classes of subgroups — too many to draw. Only its 208 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,e^2=hi,f^2=gh,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fgh,g^b=gi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265714, #72265719, #72265725
maximal normal subgroup of	#72265714, #72265719, #72265725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390505);