#6544708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1309 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451904);