#6343479

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62773	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250675);