#6578176

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#32904	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2123 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)(5,6,9,10)(7,11,8,12)(13,14,18,19)(15,20,23,27)(16,24,25,21)(17,22,26,28);(6,10)(11,12)(13,15,18,23)(14,20,19,27)(16,22,25,28)(17,24,26,21);(2,4)(13,16)(14,21)(15,17)(18,25)(19,24)(20,28)(22,27)(23,26);(5,7,9,8)(6,11,10,12)(14,19)(15,23)(16,24)(17,22)(21,25)(26,28);(5,8,9,7)(6,12,10,11)(13,17)(14,22)(15,16)(18,26)(19,28)(20,24)(21,27)(23,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=f,e^2=f,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=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, 6485372);