#127972

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 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)(17,18,20,26,24,29,23,40)(19,30,33,43,35,45,31,47)(21,37,36,46,39,28,22,27)(25,34,42,38,44,48,41,32);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,19,22,34,23,31,39,32,24,35,36,48,20,33,21,38)(18,25,28,43,26,42,27,45,29,44,37,47,40,41,46,30);(3,11)(7,15)(10,16)(12,14)(17,20,24,23)(18,26,29,40)(19,31,35,33)(21,36,39,22)(25,41,44,42)(27,37,46,28)(30,47,45,43)(32,48,38,34)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35168);