#498096

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ci,d^b=deh,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405292);