#106784

order	512
nilpotency class	3
isoclinism	isoclinic class of #106766
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

815 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13980);