#6343576

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	C₂²×C₄	C₂⁵
prime radical	#62827	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2783 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=hi,f^2=i,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, 6250772);