#6786403

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 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)(3,7)(4,6)(5,8)(17,18,22,25)(19,28,30,23)(20,33,34,40)(21,26,36,27)(24,39,41,35)(29,44,46,48)(31,42,32,38)(37,47,43,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,25,28)(20,29,34,46)(21,32,36,31)(24,37,41,43)(26,42,27,38)(33,48,40,44)(35,47,39,45);(2,6)(3,8)(10,13)(14,16)(17,20,21,35,22,34,36,39)(18,24,27,33,25,41,26,40)(19,29,32,47,30,46,31,45)(23,37,38,48,28,43,42,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,30)(20,34)(26,27)(31,32)(35,39)(37,43)(44,48);(10,13)(14,16)(17,21,22,36)(18,26,25,27)(19,31,30,32)(20,35,34,39)(23,38,28,42)(24,40,41,33)(29,45,46,47)(37,48,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6693599);