#6578350

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	#25791	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2147 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)(5,6,9,10)(7,11,8,12)(14,19)(20,28)(21,27)(22,24);(6,10)(11,12)(13,14)(15,22)(16,24)(17,19)(18,27)(20,23)(21,26)(25,28);(2,4)(13,15,17,16)(14,20,19,28)(18,25,26,23)(21,24,27,22);(5,7,9,8)(6,11,10,12)(15,23)(16,25)(18,26)(20,22)(21,27)(24,28);(5,8,9,7)(6,12,10,11)(13,16,17,15)(14,20,19,28)(18,23,26,25)(21,24,27,22)

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=dhi,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, 6485546);