#6343276

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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	#62759	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #3444 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2883 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)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,21,27,22,25,32,26)(19,28,30,20,24,31,29,23);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,25)(21,30)(22,24)(26,31)(27,28)(29,32);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(17,20)(18,24)(19,25)(21,31)(22,23)(26,30)(27,29)(28,32);(2,8)(6,9)(7,14)(13,15)(18,25)(20,23)(26,27)(28,31);(2,8)(3,10)(6,9)(11,16)(19,24)(20,23)(28,31)(29,30)

Polycyclic presentation:

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

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, 6250472);