#403277

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310473);