#422872

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi

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, 330068);