#6584575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573477
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	#36412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 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:

(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,17,21)(15,18,23,19)(16,20,24,28)(22,26,25,27);(6,10)(11,12)(15,22)(16,24)(18,26)(19,27)(20,28)(23,25);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15,17,23)(14,18,21,19)(16,25,24,22)(20,27,28,26);(5,8,9,7)(6,12,10,11)(15,22)(16,24)(18,26)(19,27)(20,28)(23,25);(13,15,17,23)(14,19,21,18)(16,25,24,22)(20,26,28,27)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,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, 6491771);