#6571032

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgi,c^2=fgi,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, 6478228);