#507807

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

495 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415003);