#6459260

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1546 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,24,22,28,21,27)(19,29,30,40,32,25,33,37)(20,26,34,38,36,41,35,42)(31,43,44,48,45,39,46,47);(2,6)(7,8)(18,24)(19,30)(20,34)(21,23)(25,29)(26,41)(27,28)(31,45)(32,33)(35,36)(39,48)(43,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,21,33,22,32,23,30)(18,25,27,40,28,29,24,37)(20,31,35,46,36,45,34,44)(26,39,42,48,41,43,38,47);(1,4,5,3)(2,8,6,7)(17,20,21,35,22,36,23,34)(18,26,27,42,28,41,24,38)(19,31,33,46,32,45,30,44)(25,39,40,48,29,43,37,47);(9,11,12,15)(10,14,13,16)(19,32)(25,29)(30,33)(31,45)(37,40)(39,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,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, 6366456);