#6343480

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	#62788	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2873 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(18,23,25,29)(24,30,31,32);(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)(27,30)(28,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,16,15)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)(19,26)(24,31)(27,28)(30,32);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,15,16,11)(17,20,22,21)(18,23,25,29)(19,27,26,28)(24,30,31,32)

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 6250676);