#6462711

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 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)(17,18,23,28,24,29,22,27)(19,30,32,40,33,46,21,25)(20,26,36,44,37,45,35,43)(31,41,34,42,47,48,38,39);(2,5)(6,8)(17,19,24,33)(18,25,29,40)(20,34,37,38)(21,22,32,23)(26,41,45,48)(27,30,28,46)(31,35,47,36)(39,43,42,44);(10,13)(14,16)(17,20,22,35,24,37,23,36)(18,26,27,43,29,45,28,44)(19,31,21,38,33,47,32,34)(25,39,46,48,40,42,30,41);(1,3,4,7)(2,6,5,8)(17,21,23,19,24,32,22,33)(18,25,28,30,29,40,27,46)(20,34,36,47,37,38,35,31)(26,42,44,48,45,39,43,41);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,27,29,28)(19,32,33,21)(20,35,37,36)(25,30,40,46)(26,43,45,44)(31,34,47,38)(39,41,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,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,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6369907);