#6674898

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	#60816	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1689 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)(21,25)(22,26)(23,28)(24,34)(27,37)(29,32)(30,33)(31,35)(36,43)(38,44)(39,47)(40,45)(41,46)(42,48);(1,3,6,13)(2,7,9,11)(4,12,5,14)(8,15,10,16)(17,19,23,31)(18,20,28,35)(21,29,37,24)(22,38,41,42)(25,32,27,34)(26,44,46,48)(30,36,47,40)(33,43,39,45);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20)(18,24)(19,25)(21,32)(22,39)(23,35)(26,38)(27,31)(28,29)(30,45)(33,41)(34,37)(36,48)(40,44)(42,46)(43,47);(2,9)(7,11)(8,10)(15,16)(17,21,23,37)(18,25,28,27)(19,29,31,24)(20,32,35,34)(22,36,41,40)(26,43,46,45)(30,38,47,42)(33,44,39,48);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,22)(18,26)(19,30)(20,33)(21,36)(23,41)(24,42)(25,43)(27,45)(28,46)(29,38)(31,47)(32,44)(34,48)(35,39)(37,40)

Polycyclic presentation:

2^9|b^2=fi,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, 6582094);