#473676

order	512
nilpotency class	3
isoclinism	isoclinic class of #457096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

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

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, 380872);