#6343537

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343524
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62835	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6250733);