#488809

order	512
nilpotency class	4
isoclinism	isoclinic class of #488699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#2184	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 396005);