#6684169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^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,d^c=di,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, 6591365);