#6344012

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343312
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	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2919 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)(11,12)(15,16)(17,18,21,25,22,26,20,24)(19,27,28,23,30,31,29,32);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,14)(12,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,3,6,10)(2,8,9,15)(4,11,5,12)(7,14,13,16)(17,20,22,21)(18,24,26,25)(19,28,30,29)(23,31,32,27);(3,10)(8,15)(11,12)(14,16);(1,4,6,5)(2,7,9,13)(3,11,10,12)(8,14,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251208);