#289914

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 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)(7,8)(9,12)(10,11)(13,14,17,22,19,24,18,23)(15,20,26,21,16,27,25,28);(1,2)(3,4)(5,6,7,12,10,11,9,8)(13,15,17,26,19,16,18,25)(14,20,22,21,24,27,23,28);(2,4)(5,7,10,9)(6,8,11,12)(13,16,19,15)(14,21,24,28)(17,26,18,25)(20,23,27,22);(5,8,7,6,10,12,9,11)(13,17,19,18)(14,22,24,23)(15,25,16,26)(20,28,27,21)

Polycyclic presentation:

2^9|a^2=f,b^2=efh,c^2=hi,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197110);