#490103

order	512
nilpotency class	4
isoclinism	isoclinic class of #490099
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	#62440	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

729 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)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,23,29)(19,25,33,47)(20,34,37,27)(21,26,39,48)(22,40,38,28)(24,35,44,31)(30,45,36,43)(32,42,41,46);(4,13)(8,12)(9,16)(11,14)(17,19)(18,24)(20,35)(21,30)(22,41)(23,33)(25,26)(27,46)(28,45)(29,44)(31,37)(32,38)(34,42)(36,39)(40,43)(47,48);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,19)(18,25)(20,36)(21,31)(22,32)(23,33)(24,26)(27,45)(28,46)(29,47)(30,37)(34,43)(35,39)(38,41)(40,42)(44,48);(2,7)(6,15)(8,12)(9,16)(17,20,22,21,23,37,38,39)(18,26,28,34,29,48,40,27)(19,30,32,35,33,36,41,31)(24,42,43,25,44,46,45,47)

Polycyclic presentation:

2^9|a^2=i,d^2=g,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,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397299);