#6520443

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500443
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2284 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)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,32)(26,34)(27,36)(29,35)(31,38)(33,37)(39,40)(41,43)(42,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,28,19,27,18,29)(14,20,23,25,24,35,22,36)(16,26,33,41,31,42,32,40)(21,34,38,39,37,44,30,43);(6,9)(7,11)(13,16,17,33,19,31,18,32)(14,21,23,38,24,37,22,30)(15,26,28,41,27,42,29,40)(20,34,25,39,35,44,36,43);(5,7,8,11)(6,10,9,12)(15,27)(16,31)(20,35)(21,37)(25,36)(28,29)(30,38)(32,33);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,32,31,33)(20,36,35,25)(21,38,37,30)(26,40,42,41)(34,39,44,43)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=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, 6427639);