#6532351

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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

1306 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)(3,6)(4,8)(5,7)(25,26,29,34,30,33,42,47)(27,35,38,43,36,48,41,31)(28,32,40,46,39,45,52,55)(37,49,51,54,50,56,53,44);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,37)(29,41)(30,36)(32,44)(33,43)(34,48)(35,47)(38,42)(39,50)(40,53)(45,54)(46,56)(49,55)(51,52);(2,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(27,36)(28,39)(31,43)(32,45)(35,48)(38,41)(40,52)(46,55);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,39)(26,32,33,45)(27,37,36,50)(29,40,42,52)(31,44,43,54)(34,46,47,55)(35,49,48,56)(38,51,41,53);(1,4,5,3)(2,8,7,6)(10,16)(14,17)(15,24)(22,23)(26,33)(27,36)(28,39)(34,47)(38,41)(40,52)(44,54)(49,56)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439547);