#416579

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1938 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323775);