#490162

order	512
nilpotency class	4
isoclinism	isoclinic class of #490087
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	#62474	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

669 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)(17,18,23,28)(19,29,32,42)(20,25,22,38)(21,34,37,45)(24,40,44,31)(26,36,27,33)(30,46,39,47)(35,41,48,43);(1,3,7,8)(2,4,11,16)(5,14,6,13)(9,12,10,15)(17,19,23,32)(18,24,28,44)(20,30,22,39)(21,35,37,48)(25,41,38,43)(26,46,27,47)(29,45,42,34)(31,33,40,36);(1,4)(2,8)(3,11)(5,15)(6,12)(7,16)(9,14)(10,13)(17,20,23,22)(18,25,28,38)(19,30,32,39)(21,33,37,36)(24,41,44,43)(26,45,27,34)(29,46,42,47)(31,35,40,48);(1,5,7,6)(2,9,11,10)(3,13,8,14)(4,15,16,12)(17,21,20,33,23,37,22,36)(18,26,38,34,28,27,25,45)(19,31,39,48,32,40,30,35)(24,42,41,47,44,29,43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dghi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397358);