#430769

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fi,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,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, 337965);