#211170

order	512
nilpotency class	3
isoclinism	isoclinic class of #19254
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

748 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118366);