#6343477

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	#62793	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2935 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(18,23,25,29)(24,30,32,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,32)(28,30);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,16,15)(17,19,22,27)(18,24,25,32)(20,28,21,26)(23,30,29,31);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)(17,20,22,21)(18,23,25,29)(19,26,27,28)(24,31,32,30);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,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, 6250673);