#107017

order	512
nilpotency class	3
isoclinism	isoclinic class of #106990
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

800 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=h,f^2=hi,g^2=hi,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,e^c=eh

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, 14213);