#6343257

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62793	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2873 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,e^b=ei,f^b=fh,d^c=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250453);