#127984

order	512
nilpotency class	4
isoclinism	isoclinic class of #127645
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8,6,9,4,13)(3,10,12,7,11,16,14,15)(17,18,20,25,23,29,37,28)(19,30,31,43,35,46,34,48)(21,38,36,26,40,27,22,41)(24,33,42,32,44,47,45,39);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,19,22,33,37,31,40,32,23,35,36,47,20,34,21,39)(18,24,27,43,25,45,41,30,29,44,38,48,28,42,26,46);(3,11)(7,15)(10,16)(12,14)(17,20,23,37)(18,25,29,28)(19,31,35,34)(21,36,40,22)(24,42,44,45)(26,27,41,38)(30,43,46,48)(32,47,39,33)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ch,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35180);