#129523

order	512
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#578	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,5,4,6,7,8)(9,10,15,20,16,12,14,19)(11,21,24,38,26,39,25,35)(13,28,31,27,32,36,30,18)(17,33,34,23,22,29,37,40);(9,11)(10,17)(12,22)(13,29)(14,24)(15,25)(16,26)(18,35)(19,34)(20,37)(21,36)(23,31)(27,38)(28,39)(30,40)(32,33);(9,12,16,10)(11,22,26,17)(13,27,32,18)(14,19,15,20)(21,23,39,40)(24,34,25,37)(28,30,36,31)(29,38,33,35)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dgi,f^b=fi,g^b=gi,e^c=ei,f^c=fi

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, 36719);