#422942

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 330138);