#128189

order	512
nilpotency class	4
isoclinism	isoclinic class of #128057
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,c^b=ch,d^b=dh,e^b=egi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277922, #72277968, #72278179
maximal normal subgroup of	#72277922, #72277968, #72278179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35385);