#6579085

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

1981 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(14,16)(15,23)(21,26)(22,25);(5,7,9,8)(6,10,12,11)(13,14)(15,24)(16,18)(17,23)(19,25)(20,26)(21,27)(22,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15,18,23)(14,20,16,27)(17,25,24,22)(19,21,28,26);(5,7,9,8)(6,11,12,10)(13,16,18,14)(15,17,23,24)(19,22,28,25)(20,26,27,21);(5,8,9,7)(6,10,12,11)(13,17)(14,21)(15,25)(16,26)(18,24)(19,27)(20,28)(22,23)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486281);