#411616

order	512
nilpotency class	3
isoclinism	isoclinic class of #393900
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=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, 318812);