#483883

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#466	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dh,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321740, #72321751, #72321835, …
maximal normal subgroup of	#72321740, #72321751, #72321835, …
maximal quotient of	#72343992

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391079);