#488534

order	512
nilpotency class	4
isoclinism	isoclinic class of #488531
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	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395730);