#6673811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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	#3005	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,31)(20,36)(21,30)(22,28)(23,29)(24,27)(25,34)(26,38)(32,48)(33,43)(35,45)(37,42)(39,44)(40,47)(41,46);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,25)(20,37)(21,33)(22,41)(23,34)(24,35)(26,28)(27,43)(29,31)(30,45)(32,38)(36,47)(39,40)(42,44)(46,48);(17,20,23,39)(18,26,29,46)(19,32,34,22)(21,38,24,41)(25,42,31,47)(27,36,30,44)(28,45,48,43)(33,40,35,37);(2,7)(6,9)(8,15)(14,16)(17,21,23,24)(18,27,29,30)(19,33,34,35)(20,38,39,41)(22,37,32,40)(25,43,31,45)(26,36,46,44)(28,47,48,42);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22)(18,28)(19,20)(21,40)(23,32)(24,37)(25,44)(26,45)(27,42)(29,48)(30,47)(31,36)(33,41)(34,39)(35,38)(43,46)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=i,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,g^c=gi,h^c=hi,e^d=ei,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, 6581007);