#134574

order	512
nilpotency class	4
isoclinism	isoclinic class of #124874
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1560
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8585
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

376 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=h,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^c=eh,f^c=fghi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41770);