#206323

order	512
nilpotency class	3
isoclinism	isoclinic class of #19745
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1088 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)(6,9)(7,11)(10,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,27)(26,28);(2,4)(5,6,7,9,8,10,11,12)(14,18)(19,26)(20,25)(21,27);(13,15,17,23,16,22,24,28)(14,19,20,21,18,26,25,27);(6,10)(9,12)(15,22)(19,26)(21,27)(23,28)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=f,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113519);