#428649

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1101 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335845);