#6579056

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573477
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	#36517	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,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=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, 6486252);