#6520309

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427505);