#202761

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

782 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)(5,6,8,11)(7,9,12,10)(13,14,19,21)(15,22,25,28)(16,18,26,17)(20,23,27,24);(1,3,4,2)(13,15,18,24,19,25,17,23)(14,20,16,22,21,27,26,28);(6,9)(7,12)(10,11)(13,16,18,21,19,26,17,14)(15,22,24,27,25,28,23,20);(13,17,19,18)(14,16,21,26)(15,23,25,24)(20,22,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109957);