#448368

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1215 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=ei,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,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, 355564);