#423669

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 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)(11,23,28,36)(12,29,31,38)(13,32,34,19)(14,21,15,20)(17,27,37,26)(18,30,39,25)(24,35,33,40);(5,6)(7,8)(9,11)(10,17)(12,13)(14,27)(15,26)(16,28)(18,19)(20,36)(21,23)(22,37)(24,25)(29,35)(30,33)(31,34)(32,39)(38,40);(1,2,3,4)(6,8)(9,12,15,25,16,31,14,30)(10,18,21,29,22,39,20,38)(11,24,27,13,28,33,26,34)(17,35,36,19,37,40,23,32);(9,13,14,24,16,34,15,33)(10,19,20,35,22,32,21,40)(11,25,26,12,28,30,27,31)(17,29,23,18,37,38,36,39)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,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, 330865);