#422893

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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	#23426	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1933 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)(9,10,16,22)(11,23,26,18)(12,17,25,34)(13,28,32,36)(14,21,15,20)(19,30,37,31)(24,39,40,33)(27,38,29,35);(1,2,3,4)(5,6)(7,8)(9,11,15,25,16,26,14,12)(10,17,21,18,22,34,20,23)(13,29,31,24,32,27,30,40)(19,35,28,33,37,38,36,39);(6,8)(9,12,14,26,16,25,15,11)(10,18,20,17,22,23,21,34)(13,27,31,40,32,29,30,24)(19,35,28,33,37,38,36,39);(2,4)(9,13)(10,19)(11,24)(12,27)(14,31)(15,30)(16,32)(17,33)(18,35)(20,28)(21,36)(22,37)(23,38)(25,29)(26,40)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330089);