#6579288

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486484);