#6344113

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
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	#62800	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3008 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,f^2=i,g^2=hi,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=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, 6251309);