#482353

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

767 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)(4,8)(5,7)(10,15,16,13)(12,14)(17,18,21,27,23,25,22,37)(19,28,30,45,31,44,38,43)(20,32,34,42,36,47,35,26)(24,33,40,46,39,48,41,29);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,39)(26,43)(27,40)(28,32)(29,35)(34,46)(36,48)(37,41)(42,45)(44,47);(2,7)(6,9)(8,14)(13,15);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(18,25)(24,39)(26,42)(27,37)(28,44)(32,47)(40,41)(43,45)

Polycyclic presentation:

2^9|a^2=f,d^2=ghi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cghi,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, 389549);