#6579050

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	#36083	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2010 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)(14,19)(15,21)(20,23)(22,28);(5,7,9,8)(6,10,12,11)(13,14)(15,18)(16,22)(17,19)(20,25)(21,26)(23,27)(24,28);(2,4)(5,7,9,8)(6,11,12,10)(14,15)(18,26)(19,21)(20,22)(23,28)(25,27);(5,7,9,8)(6,11,12,10)(13,15,17,21)(14,18,19,26)(16,23,24,20)(22,27,28,25);(5,8,9,7)(6,10,12,11)(13,16,17,24)(14,20,19,23)(15,22,21,28)(18,27,26,25)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6486246);