#467126

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1207 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 374322);