#6578205

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2129 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)(15,22)(19,23)(24,25)(27,28);(6,10)(11,12)(13,14,17,21)(15,23,22,19)(16,18,26,20)(24,28,25,27);(2,4)(14,18)(15,24)(16,26)(20,21)(22,25)(27,28);(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,24)(17,22)(18,27)(20,28)(21,23)(25,26);(5,8,9,7)(6,12,10,11)(14,20)(15,24)(16,26)(18,21)(19,23)(22,25)

Polycyclic presentation:

2^9|a^2=f,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=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, 6485401);