#289478

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

718 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,c^2=i,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 196674);