#443556

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1640 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)(9,10,16,22)(11,17,26,37)(12,27,30,34)(13,31,32,38)(14,21,15,20)(18,29,39,23)(19,33,40,28)(24,36,25,35);(1,2,5,6)(3,7,4,8)(9,11,16,26)(10,17,22,37)(12,28,30,33)(13,23,32,29)(14,24,15,25)(18,38,39,31)(19,34,40,27)(20,35,21,36);(1,3,5,4)(2,7,6,8)(9,12,14,29,16,30,15,23)(10,18,21,27,22,39,20,34)(11,13,25,28,26,32,24,33)(17,19,35,38,37,40,36,31);(1,4,5,3)(2,7,6,8)(9,13)(10,19)(11,23)(12,25)(14,28)(15,33)(16,32)(17,34)(18,35)(20,31)(21,38)(22,40)(24,30)(26,29)(27,37)(36,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=eg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,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, 350752);