#6576486

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2755 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)(7,12)(8,10)(9,11)(13,14,17,18)(15,19,22,26)(16,20,25,27)(21,24,28,23);(5,7,9,8)(6,10,11,12)(14,18)(15,19)(20,27)(21,24)(22,26)(23,28);(2,4)(13,15,17,22)(14,19,18,26)(16,23,25,24)(20,21,27,28);(5,8,9,7)(6,10,11,12)(13,16,17,25)(14,20,18,27)(15,21,22,28)(19,24,26,23);(13,15)(14,19)(16,24)(17,22)(18,26)(20,28)(21,27)(23,25)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,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, 6483682);