#6673808

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

1614 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)(19,25)(20,32)(21,27)(22,30)(23,28)(24,29)(26,34)(31,41)(33,47)(35,39)(36,43)(37,44)(38,46)(40,45)(42,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,24,32)(18,25,29,20)(21,36,37,42)(22,26,23,39)(27,43,44,48)(28,35,30,34)(31,40,47,38)(33,46,41,45);(17,20,24,25)(18,26,29,39)(19,30,32,28)(21,33,37,41)(22,35,23,34)(27,42,44,36)(31,46,47,45)(38,43,40,48);(2,7)(6,9)(8,15)(14,16)(21,37)(27,44)(31,47)(33,41)(36,42)(38,40)(43,48)(45,46);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,27)(19,31)(20,33)(22,38)(23,40)(24,37)(25,41)(26,42)(28,45)(29,44)(30,46)(32,47)(34,48)(35,43)(36,39)

Polycyclic presentation:

2^9|b^2=fi,c^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,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, 6581004);