#6464965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

1375 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)(17,18,23,27,24,28,22,38)(19,29,31,41,32,40,20,25)(21,26,36,44,37,45,35,47)(30,42,34,43,46,48,33,39);(2,5)(6,8)(17,19,24,32)(18,25,28,41)(20,22,31,23)(21,34,37,33)(26,42,45,48)(27,40,38,29)(30,35,46,36)(39,47,43,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,19,24,31,22,32)(18,25,27,29,28,41,38,40)(21,34,36,46,37,33,35,30)(26,43,44,48,45,39,47,42);(17,21,22,35,24,37,23,36)(18,26,38,47,28,45,27,44)(19,30,20,33,32,46,31,34)(25,39,40,48,41,43,29,42);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,27,28,38)(19,31,32,20)(21,35,37,36)(25,40,41,29)(26,44,45,47)(30,34,46,33)(39,48,43,42)

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,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372161);