#491190

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
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	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

396 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)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18)(19,28)(20,25)(21,35)(22,34)(23,27)(24,31)(26,36)(29,41)(30,42)(32,48)(33,44)(37,38)(39,45)(40,47)(43,46);(17,19,29,45,22,32,21,31,23,33,36,46,20,30,37,47)(18,24,38,48,25,39,26,28,27,40,35,42,34,43,41,44);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,18)(19,24)(20,34)(21,26)(22,25)(23,27)(28,31)(29,38)(30,43)(32,39)(33,40)(35,36)(37,41)(42,46)(44,47)(45,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,20,23,22)(18,25,27,34)(19,30,33,32)(21,29,37,36)(24,39,40,43)(26,35,41,38)(28,42,44,48)(31,45,47,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398386);