#491383

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

436 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)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(18,23)(20,31)(21,32)(24,33)(25,34)(26,35)(27,38)(29,44)(30,43)(36,37)(39,42)(40,48)(41,47)(45,46);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,18,24,38,21,27,20,23,22,26,31,37,32,36,33,35)(19,25,39,48,30,40,29,34,28,41,44,46,43,45,42,47);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,34)(24,39)(26,41)(27,40)(31,44)(32,43)(33,42)(35,47)(36,45)(37,46)(38,48);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(18,26)(19,28)(23,35)(27,36)(29,42)(30,43)(37,38)(39,44)

Polycyclic presentation:

2^9|b^2=egi,d^2=h,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=dhi,d^c=di,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398579);