#491420

order	512
nilpotency class	4
isoclinism	isoclinic class of #491245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19222	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

400 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=egi,d^2=hi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=dh,d^c=di,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, 398616);