#105280

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	C₄×C₄⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ci,d^a=dh,f^a=fh,c^b=ch

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12476);