#205289

order	512
nilpotency class	3
isoclinism	isoclinic class of #19279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1028 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fi,d^2=hi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112485);