#404630

order	512
nilpotency class	3
isoclinism	isoclinic class of #404630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14896
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311826);