#491145

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	#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

404 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,23)(20,32)(21,33)(24,34)(25,35)(26,39)(27,38)(28,43)(30,41)(31,46)(36,37)(40,45)(42,48)(44,47);(17,18,24,39,21,26,20,23,22,27,32,37,33,36,34,38)(19,28,40,47,31,44,30,43,29,25,41,48,46,42,45,35);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,25)(20,30)(21,31)(22,29)(23,35)(24,40)(26,42)(27,28)(32,41)(33,46)(34,45)(36,44)(37,47)(38,43)(39,48);(2,7)(6,15)(8,12)(9,16)(19,29)(25,28)(30,45)(31,46)(35,43)(40,41)(42,44)(47,48)

Polycyclic presentation:

2^9|b^2=egi,c^2=h,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,c^b=ci,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, 398341);