#135371

order	512
nilpotency class	4
isoclinism	isoclinic class of #124974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1517
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8716
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

327 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,13)(4,8)(5,19,21,26)(6,10)(7,12)(9,15,25,31)(11,18,28,22)(14,29)(16,27)(17,30)(20,24,32,23);(2,8,12,10)(9,23,25,24)(11,19,28,26)(13,29,30,27);(1,3,4,14,7,17,6,16)(2,9,10,24,12,25,8,23)(5,20,18,31,21,32,22,15)(11,27,26,30,28,29,19,13)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=d,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,e^c=eh,f^c=fgh,g^c=gi,h^c=hi,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, 42567);