#107927

order	512
nilpotency class	3
isoclinism	isoclinic class of #107918
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×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

853 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15123);