#107849

order	512
nilpotency class	3
isoclinism	isoclinic class of #9519
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15045);