#409835

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=f,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317031);