#6456977

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1453 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(18,24)(19,27)(26,32)(29,30);(1,3,5,4)(2,6,8,7)(17,18,22,24)(19,28,27,31)(20,30,25,29)(21,26,23,32);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19)(18,25)(20,24)(21,30)(22,27)(23,29)(26,31)(28,32);(1,4,5,3)(2,6,8,7)(17,20,22,25)(18,19,24,27)(21,31,23,28)(26,29,32,30);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,24,32)(19,29,27,30)(20,31,25,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,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, 6364173);