#404525

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1108 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)(3,7,8,5)(18,19)(20,27,28,30)(23,31,25,32)(24,26);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,19)(20,23,28,25)(22,26,29,24)(27,31,30,32);(10,13)(14,16)(17,19,21,18)(20,25,28,23)(22,24,29,26)(27,32,30,31);(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,30)(24,31)(26,32)(27,29)

Polycyclic presentation:

2^9|a^2=ei,b^2=efi,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi,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, 311721);