#6454040

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	#35189	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3126 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,24,26,27)(16,21,19,20)(17,25,28,23);(2,4)(13,15)(14,19)(16,22)(17,20)(18,26)(21,28)(23,27)(24,25);(6,9)(10,12)(13,16,18,19)(14,20,22,21)(15,23,26,25)(17,24,28,27);(13,16,18,19)(14,21,22,20)(15,25,26,23)(17,24,28,27);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(20,21)(23,25)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361236);