#6441251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348447);