#477201

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384397);