#127684

order	512
nilpotency class	4
isoclinism	isoclinic class of #127645
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	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

422 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,21,25,22,27,37,26)(19,28,31,46,29,45,32,42)(20,33,35,24,36,43,48,44)(23,38,40,30,39,34,41,47);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19,21,32,22,29,37,31)(18,23,26,40,27,39,25,41)(20,34,35,30,36,38,48,47)(24,42,33,28,44,46,43,45);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(19,29)(23,39)(28,45)(30,47)(31,32)(34,38)(40,41)(42,46)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=cgh,d^b=dh,e^b=ei,g^b=gi,h^b=hi

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, 34880);