#398041

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305237);