#6674249

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	#60763	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(19,29)(20,27)(21,26)(22,25)(23,28)(24,32)(30,40)(31,45)(33,39)(34,48)(35,42)(36,41)(37,44)(38,47)(43,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,21,32)(18,24,26,29)(20,34,35,46)(22,33,23,30)(25,40,28,39)(27,43,42,48)(31,36,47,37)(38,41,45,44);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,25)(19,30)(20,35)(22,23)(26,28)(27,44)(31,34)(32,33)(38,45)(39,40)(41,42)(46,47);(2,9)(7,11)(8,10)(15,16)(18,26)(24,29)(25,28)(27,42)(38,45)(39,40)(41,44)(43,48);(1,5,6,4)(2,10,9,8)(3,12,13,14)(7,15,11,16)(17,20)(18,27)(19,31)(21,35)(22,36)(23,37)(24,38)(25,41)(26,42)(28,44)(29,45)(30,46)(32,47)(33,34)(39,48)(40,43)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,e^2=fi,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, 6581445);