#282081

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

765 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=h,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189277);