#6596592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38914	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2961 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)(13,14,20,23,19,16,18,15)(17,24,28,22,26,21,27,25);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,23)(14,20,16,18)(17,25,26,22)(21,27,24,28);(6,9)(7,11)(13,16,19,14)(15,20,23,18)(17,21,26,24)(22,27,25,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,22)(16,24)(18,27)(19,26)(20,28)(23,25);(6,9)(7,11)(17,26)(21,24)(22,25)(27,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6503788);