#6623941

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(17,18,22,28,23,26,21,27)(19,29,33,24,31,40,32,39)(20,34,36,41,37,42,35,25)(30,44,47,48,45,38,46,43);(2,5)(6,8)(10,14)(12,15)(17,19,23,31)(18,24,26,39)(20,30,37,45)(21,33,22,32)(25,38,41,44)(27,40,28,29)(34,48,42,43)(35,47,36,46);(9,10)(11,12)(13,14)(15,16)(17,20,23,37)(18,25,26,41)(19,30,31,45)(21,36,22,35)(24,38,39,44)(27,34,28,42)(29,43,40,48)(32,47,33,46);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,15,14,12)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(9,12,13,15)(10,11,14,16)(18,26)(24,39)(25,41)(27,28)(29,40)(34,42)(38,44)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6531137);