#127872

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gi,e^2=h,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35068);