#6441119

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1760 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=f,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6348315);