#6630853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,29,31,40)(20,34,36,42)(21,28,23,27)(24,33,38,32)(26,35,41,37)(30,39,44,48)(43,45,47,46);(2,5)(6,8)(17,19,23,33,22,31,21,32)(18,24,28,40,25,38,27,29)(20,30,37,46,36,44,35,45)(26,39,34,47,41,48,42,43);(9,11,12,16)(10,13,14,15)(18,25)(24,38)(26,41)(27,28)(29,40)(34,42)(39,48)(43,47);(1,3,4,7)(2,6,5,8)(17,20,21,35,22,36,23,37)(18,26,27,42,25,41,28,34)(19,30,32,45,31,44,33,46)(24,39,29,43,38,48,40,47);(19,31)(24,38)(29,40)(30,44)(32,33)(39,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538049);