#470302

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1201 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377498);