#385748

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi

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, 292944);