#127979

order	512
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

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	#102
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

428 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)(19,29)(20,34)(21,25)(22,39)(23,28)(24,31)(26,37)(27,36)(30,43)(32,42)(33,48)(35,40)(38,44)(41,47)(45,46);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,19,21,32,22,33,37,31,23,30,38,46,36,47,20,35)(18,24,26,41,27,42,25,43,28,40,34,48,39,45,44,29);(3,11)(7,15)(10,16)(12,14)(19,30)(24,40)(29,43)(31,35)(32,46)(33,47)(41,48)(42,45)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gi,e^2=hi,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, 35175);