#490149

order	512
nilpotency class	4
isoclinism	isoclinic class of #490075
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397345);