#6620248

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6527444);