#6344240

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343554
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	C₄²	C₂⁵
prime radical	#62834	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2973 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251436);