#107113

order	512
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

782 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:

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

Polycyclic presentation:

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

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, 14309);