#135478

order	512
nilpotency class	4
isoclinism	isoclinic class of #135477
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

548 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=ei,f^c=fghi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42674);