#6578334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#25787	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2103 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)(5,6,9,10)(7,11,8,12)(13,14,19,16)(15,20,24,25)(17,21,28,26)(18,22,23,27);(6,10)(11,12)(13,15)(14,20)(16,25)(17,23)(18,28)(19,24)(21,27)(22,26);(2,4)(13,16,19,14)(15,21,24,26)(17,20,28,25)(18,22,23,27);(5,7,9,8)(6,11,10,12)(13,17,19,28)(14,21,16,26)(15,23,24,18)(20,27,25,22);(5,8,9,7)(6,12,10,11)(15,17)(18,23)(20,21)(22,27)(24,28)(25,26)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485530);