#126837

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

416 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:

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

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=fg,g^a=gi,c^b=ch,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34033);