#6465042

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=hi,e^2=g,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,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, 6372238);