#490050

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
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	#61477	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

663 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397246);