#410833

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=i,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^c=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, 318029);