#6343264

order	512
nilpotency class	3
isoclinism	isoclinic class of #62283
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

2836 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)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,21,26,22,25,32,38)(19,23,29,34,28,35,41,45)(20,24,31,37,30,36,42,46)(27,33,40,44,39,43,47,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,35)(26,34)(30,39)(31,40)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,36)(26,37)(28,39)(29,40)(32,42)(34,44)(35,43)(38,46)(41,47)(45,48);(2,8)(6,9)(7,14)(13,15)(20,30)(24,36)(27,39)(31,42)(33,43)(37,46)(40,47)(44,48);(3,10)(7,14)(11,16)(13,15)(18,25)(19,28)(24,36)(26,38)(27,39)(29,41)(37,46)(40,47)

Polycyclic presentation:

2^9|a^2=g,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=di,e^c=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, 6250460);