#282327

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189523);