#449023

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1017 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:

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

Polycyclic presentation:

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

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, 356219);