#6464855

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1384 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)(17,18,22,26,23,27,32,40)(19,28,30,38,31,43,20,24)(21,25,36,42,35,41,44,47)(29,39,34,46,45,48,33,37);(2,5)(6,8)(17,19,23,31)(18,24,27,38)(20,32,30,22)(21,34,35,33)(25,39,41,48)(26,43,40,28)(29,44,45,36)(37,47,46,42);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,19,23,30,32,31)(18,24,26,28,27,38,40,43)(21,33,36,29,35,34,44,45)(25,37,42,39,41,46,47,48);(17,21,32,44,23,35,22,36)(18,25,40,47,27,41,26,42)(19,29,20,33,31,45,30,34)(24,37,43,48,38,46,28,39);(9,11,12,15)(10,14,13,16)(21,35)(25,41)(29,45)(33,34)(36,44)(37,46)(39,48)(42,47)

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,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, 6372051);