#6448561

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(18,22)(19,25)(24,29)(27,30);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,26)(20,22)(21,29)(23,27)(24,28)(25,32)(30,31);(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,25)(21,30)(22,31)(24,32)(26,29)(27,28);(18,24)(21,28)(22,29)(23,32)(26,31)(27,30);(1,3,4,7)(2,6,5,8)(18,22)(19,25)(24,29)(27,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,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, 6355757);