#483920

order	512
nilpotency class	4
isoclinism	isoclinic class of #483919
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	C₂³	#442
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ 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:

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

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,c^b=ch,e^b=egi,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287124, #72287142, #72287162, …
maximal normal subgroup of	#72287124, #72287142, #72287162, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391116);