#482211

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,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, 389407);