#419917

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1590 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,h^c=hi,f^d=fi,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, 327113);