#6454087

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3152 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)(7,10)(8,9)(11,12)(13,14,20,17)(15,24,22,16)(18,23,19,26)(21,28,27,25);(2,4)(13,15,20,22)(14,21,17,27)(16,26,24,23)(18,25,19,28);(6,9)(10,12)(13,16)(14,22)(15,17)(18,21)(19,27)(20,24)(23,25)(26,28);(13,17,20,14)(15,24,22,16)(18,26,19,23)(21,28,27,25);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,26)(15,25,22,28)(16,27,24,21)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=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, 6361283);