#390444

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1369 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=egh,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 297640);