#289932

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

689 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)(8,18)(9,19)(10,14)(11,17)(13,15)(16,20)(21,22,26,28,25,30,24,29)(23,27,31,36,32,34,33,35);(1,2)(3,4)(5,7,15,8,10,16,9,17)(6,11,19,20,14,18,13,12)(21,23,24,33,25,32,26,31)(22,27,29,35,30,34,28,36);(2,4)(7,16)(8,17)(11,18)(12,20)(22,28)(23,31)(24,26)(27,34)(29,30)(32,33);(5,8,15,16,10,17,9,7)(6,12,19,11,14,20,13,18)(23,32)(27,34)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,d^2=eh,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=dhi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197128);