#490334

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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

670 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=ghi,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, 397530);