#491024

order	512
nilpotency class	4
isoclinism	isoclinic class of #491017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33817	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

449 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)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18,21,24)(19,29,31,46)(20,28,23,26)(22,35,34,43)(25,41,39,32)(27,33,40,37)(30,47,38,44)(36,45,48,42);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(18,24)(25,39)(26,28)(27,40)(29,46)(35,43)(42,45)(44,47);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,25)(20,30)(21,31)(22,36)(23,38)(24,39)(26,42)(27,44)(28,45)(29,35)(32,37)(33,41)(34,48)(40,47)(43,46);(2,8)(3,9)(6,13)(10,15)(17,20,21,23)(18,26,24,28)(19,30,31,38)(22,33,34,37)(25,42,39,45)(27,43,40,35)(29,44,46,47)(32,36,41,48)

Polycyclic presentation:

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

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, 398220);