#470364

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1061 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=e,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=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, 377560);