#6578870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2014 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)(7,11)(8,10)(9,12)(13,14,18,16)(15,22,17,20)(19,21,28,26)(23,25,27,24);(5,7,9,8)(6,10,12,11)(13,15,18,17)(14,20,16,22)(19,24,28,25)(21,27,26,23);(2,4)(5,7,9,8)(6,11,12,10)(13,16,18,14)(15,23,17,27)(19,21,28,26)(20,24,22,25);(5,7,9,8)(6,11,12,10)(13,17)(14,20)(15,18)(16,22)(19,25)(21,27)(23,26)(24,28);(5,8,9,7)(6,10,12,11)(15,24)(17,25)(19,28)(20,27)(21,26)(22,23)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,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, 6486066);