#127261

order	512
nilpotency class	4
isoclinism	isoclinic class of #127249
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

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 ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

410 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)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,27,24,29,23,39)(19,30,33,46,31,45,37,42)(20,28,22,38,36,44,35,26)(25,34,41,32,40,48,43,47);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,34,21,37,36,47,24,31,35,48,23,33,20,32)(18,25,28,42,39,41,26,30,29,40,44,46,27,43,38,45);(19,31)(25,40)(30,45)(32,47)(33,37)(34,48)(41,43)(42,46)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ghi,e^2=i,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ci,e^b=eh

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, 34457);