#282009

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
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

725 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)(5,6,10,7,9,13,11,14)(8,16,19,17,18,15,20,12)(21,22,26,28,25,30,24,29)(23,27,32,35,31,36,33,34);(2,4)(5,7,10,13,9,14,11,6)(8,17,19,15,18,12,20,16)(21,23,26,32,25,31,24,33)(22,27,28,35,30,36,29,34);(5,8,9,18)(6,12,13,17)(7,15,14,16)(10,20,11,19)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(2,4)(6,13)(7,14)(12,17)(15,16)(21,24,25,26)(22,29,30,28)(23,32,31,33)(27,35,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189205);