#6497649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404845);