#219250

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,d^c=dhi

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, 126446);