#6674244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60766	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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)(3,11)(4,8)(5,10)(6,9)(7,13)(12,16)(14,15)(17,18)(19,20)(21,26)(22,27)(23,29)(24,28)(25,33)(30,34)(31,35)(32,36)(37,42)(38,43)(39,47)(40,45)(41,46)(44,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,20)(21,25)(22,38)(23,31)(24,32)(26,33)(27,43)(28,36)(29,35)(30,40)(34,45)(37,47)(39,42)(41,44)(46,48);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20)(18,25)(19,29)(21,33)(22,39)(23,35)(24,36)(26,32)(27,44)(28,31)(30,42)(34,41)(37,48)(38,46)(40,43)(45,47);(2,9)(7,11)(8,10)(15,16)(17,21,24,23)(18,26,28,29)(19,25,32,31)(20,33,36,35)(22,37,41,40)(27,42,46,45)(30,38,47,44)(34,43,39,48);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,22)(18,27)(19,30)(20,34)(21,37)(23,40)(24,41)(25,38)(26,42)(28,46)(29,45)(31,44)(32,47)(33,43)(35,48)(36,39)

Polycyclic presentation:

2^9|c^2=f,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581440);