#402989

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ei,c^2=ei,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,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, 310185);