#404863

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1376 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)(3,8)(4,6)(5,7)(17,18,22,26)(19,27,20,29)(21,31,32,25)(23,30,24,28);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,20)(18,23,26,24)(21,30,32,28)(25,29,31,27);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,19)(18,24,26,23)(21,28,32,30)(25,27,31,29);(9,11,12,15)(10,14,13,16)(21,32)(25,31)(27,29)(28,30)

Polycyclic presentation:

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

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, 312059);