#6571004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,33,30,34,44,50)(27,35,39,46,37,47,38,31)(28,40,42,49,43,48,41,32)(36,45,53,56,52,55,54,51);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,30,37)(26,31,34,46)(28,36,43,52)(29,38,44,39)(32,45,49,55)(33,47,50,35)(40,51,48,56)(41,53,42,54);(1,3,5,4)(2,7,8,6)(10,15)(11,18)(16,22)(19,24)(25,28,30,43)(26,32,34,49)(27,36,37,52)(29,41,44,42)(31,45,46,55)(33,48,50,40)(35,51,47,56)(38,53,39,54);(1,4,5,3)(2,6,8,7)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(27,37)(31,46)(35,47)(36,52)(38,39)(45,55)(51,56)(53,54);(12,20)(16,22)(19,24)(21,23)(25,29,30,44)(26,33,34,50)(27,38,37,39)(28,41,43,42)(31,47,46,35)(32,48,49,40)(36,53,52,54)(45,56,55,51)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fg,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,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, 6478200);