#6484608

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473165
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,27)(20,31)(21,36)(22,25)(23,29)(24,35)(26,33)(28,42)(30,38)(32,47)(34,46)(37,48)(39,45)(40,43)(41,44);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,21,30,22,29,33,44)(18,23,26,38,25,27,36,41)(20,32,35,48,34,28,43,45)(24,39,31,42,40,37,46,47);(3,9)(6,13)(10,11)(14,15)(17,20,33,43,22,34,21,35)(18,24,36,46,25,40,26,31)(19,28,44,48,29,32,30,45)(23,37,41,42,27,39,38,47);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(19,29)(20,34)(23,27)(24,40)(30,44)(31,46)(35,43)(38,41);(2,8)(6,13)(7,16)(14,15)(18,25)(20,34)(23,27)(26,36)(28,32)(35,43)(38,41)(45,48)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391804);