#107867

order	512
nilpotency class	3
isoclinism	isoclinic class of #107858
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

753 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:

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

Polycyclic presentation:

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

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, 15063);