#6576406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1977 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483602);