#6579229

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2003 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,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, 6486425);