#6579125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1965 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:

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

Polycyclic presentation:

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

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, 6486321);