#483890

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	#2186	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

426 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)(17,18,22,28,23,29,20,26)(19,30,31,44,34,45,33,25)(21,36,39,47,40,46,35,27)(24,41,42,32,43,37,48,38);(17,19)(18,24)(20,31)(21,37)(22,33)(23,34)(25,27)(26,42)(28,48)(29,43)(30,46)(32,39)(35,38)(36,45)(40,41)(44,47);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25,29,44)(20,33)(21,38)(22,31)(23,34)(24,27,43,47)(26,45,28,30)(32,40)(35,37)(36,48,46,42)(39,41);(2,7)(6,12)(8,14)(13,16)(17,20,23,22)(18,26,29,28)(19,31,34,33)(21,35,40,39)(24,42,43,48)(25,30,44,45)(27,46,47,36)(32,37,38,41)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,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	#72216788, #72216809, #72216898, …
maximal normal subgroup of	#72216788, #72216809, #72216898, …
maximal quotient of	C₂×#483890, #72318549, #72318586, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391086);