#322546

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6,9,12,8,10,16,19)(7,13,15,18,14,20,17,11)(21,22,24,30,26,29,25,28)(23,27,31,34,32,36,33,35);(1,2,4,3)(6,10)(11,18)(12,19)(13,20)(21,23,24,31,26,32,25,33)(22,27,30,34,29,36,28,35);(5,7,8,14)(6,11,10,18)(9,17,16,15)(12,20,19,13)(22,28)(23,31)(24,25)(29,30)(32,33)(34,35);(1,3,4,2)(22,29)(23,32)(28,30)(31,33)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=hi,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,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, 229742);