#489634

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
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	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

641 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(18,25)(19,31)(20,35)(21,37)(22,23)(26,41)(27,30)(28,38)(29,36)(32,33)(34,39)(40,43)(42,45)(44,48);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,18,24,30)(19,26,34,44)(20,36,21,38)(22,29,23,28)(25,35,27,37)(31,45,39,43)(32,41,33,48)(40,47,42,46);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,24,34)(18,26,30,44)(20,33,21,32)(22,39,23,31)(25,40,27,42)(28,45,29,43)(35,47,37,46)(36,48,38,41);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,37,24,21,23,35)(18,27,28,38,30,25,29,36)(19,32,39,47,34,33,31,46)(26,40,45,48,44,42,43,41)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396830);