#6462656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369852);