#6344210

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	#62836	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2865 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,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, 6251406);