#6465107

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6372303);