#289734

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196930);