#106394

order	512
nilpotency class	3
isoclinism	isoclinic class of #106380
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

850 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)(3,10)(4,13)(5,8)(6,9)(7,11)(12,16)(14,15)(17,18,21,25,22,26,20,24)(19,27,28,31,29,32,30,23);(1,3,5,11,6,12,4,14)(2,7,8,16,9,15,13,10)(17,19)(18,23,26,31)(20,28)(21,30)(22,29)(24,27,25,32);(1,2)(3,7)(4,13)(5,8)(6,9)(10,14)(11,16)(12,15)

Polycyclic presentation:

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

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, 13590);