#289926

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

729 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,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, 197122);