#281566

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 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)(5,6,9,13,11,14,10,7)(8,16,19,12,18,15,17,20)(21,22,26,28,25,30,24,29)(23,27,33,36,31,35,32,34);(2,4)(5,7,10,14,11,13,9,6)(8,12,17,16,18,20,19,15)(21,23,24,32,25,31,26,33)(22,27,29,34,30,35,28,36);(5,8)(6,12)(7,15)(9,17)(10,19)(11,18)(13,16)(14,20)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(2,4)(5,9,11,10)(6,13,14,7)(8,17,18,19)(12,16,20,15)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188762);