#105762

order	512
nilpotency class	3
isoclinism	isoclinic class of #105752
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

758 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di,f^c=fh

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, 12958);