#6584520

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
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2009 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,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, 6491716);