#6584498

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ 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:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,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, 6491694);