#135327

order	512
nilpotency class	4
isoclinism	isoclinic class of #124838
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1507
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8713
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227805, #72227810, #72227812
maximal normal subgroup of	#72227805, #72227810, #72227812

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42523);