#443554

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
prime radical	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1576 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)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,27)(22,30)(24,31)(25,34)(28,32)(29,33)(35,37)(36,39)(38,40);(1,2,5,6)(3,7,4,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,24)(22,35)(23,31)(25,34)(26,27)(28,33)(29,32)(30,37)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(9,11,13,24,15,25,14,23)(10,16,19,21,20,31,18,34)(12,22,27,37,28,38,29,36)(17,30,32,35,33,40,26,39);(1,4,5,3)(2,7,6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,26)(19,32)(20,33)(21,35)(23,36)(24,37)(25,38)(31,40)(34,39)

Polycyclic presentation:

2^9|b^2=e,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, 350750);