#508463

order	512
nilpotency class	4
isoclinism	isoclinic class of #508289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gh,h^b=hi,d^c=de,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415659);