#107172

order	512
nilpotency class	3
isoclinism	isoclinic class of #107154
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

789 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ei,d^c=di,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, 14368);