#6578564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#36004	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2113 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485760);