#6465104

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

1401 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)(17,18,23,27,24,28,22,40)(19,29,21,38,31,42,39,25)(20,32,36,48,37,26,35,45)(30,46,33,41,47,43,34,44);(2,5)(6,8)(17,19,24,31)(18,25,28,38)(20,33,37,34)(21,23,39,22)(26,43,32,46)(27,42,40,29)(30,35,47,36)(41,48,44,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,23,36,24,37,22,35)(18,26,27,45,28,32,40,48)(19,30,21,33,31,47,39,34)(25,41,29,43,38,44,42,46);(17,21,23,31,24,39,22,19)(18,25,27,29,28,38,40,42)(20,34,36,30,37,33,35,47)(26,44,45,46,32,41,48,43);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,27,28,40)(19,21,31,39)(20,35,37,36)(25,42,38,29)(26,45,32,48)(30,33,47,34)(41,46,44,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=h,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,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, 6372300);