#6343254

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	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,9,6,10,4,8)(3,11,14,7,13,16,12,15)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,43)(31,38)(32,40)(33,41)(34,42)(36,39)(37,45)(44,47)(46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,44)(33,46)(34,45)(40,43)(41,47)(42,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,20,23,33)(18,25,27,41)(19,29,31,46)(21,34,22,32)(24,37,39,47)(26,42,35,40)(28,43,38,48)(30,45,36,44);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,33,34)(24,38,39,28)(25,40,41,42)(29,44,46,45)(37,43,47,48);(3,13)(7,15)(11,16)(12,14)(18,27)(24,39)(25,41)(26,35)(28,38)(37,47)(40,42)(43,48)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^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, 6250450);