#6578565

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2051 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)(13,14,18,20)(15,21,25,17)(16,22,19,23)(24,27,28,26);(5,7,9,8)(6,10,12,11)(14,20)(15,24)(16,19)(17,27)(21,26)(25,28);(2,4)(13,15)(14,21)(16,24)(17,20)(18,25)(19,28)(22,27)(23,26);(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,25)(17,21)(18,19)(20,23);(5,8,9,7)(6,10,12,11)(13,17,18,21)(14,15,20,25)(16,26,19,27)(22,24,23,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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, 6485761);