#417853

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3012	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2770 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:

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

Polycyclic presentation:

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

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, 325049);