#483884

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	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 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)(17,18,22,26,23,28,20,34)(19,29,30,46,33,45,32,25)(21,36,39,47,40,48,35,27)(24,41,43,31,44,37,42,38);(17,19)(18,24)(20,30)(21,37)(22,32)(23,33)(25,27)(26,42)(28,44)(29,48)(31,39)(34,43)(35,38)(36,45)(40,41)(46,47);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,25,28,46)(20,32)(21,38)(22,30)(23,33)(24,27,44,47)(26,29,34,45)(31,40)(35,37)(36,42,48,43)(39,41);(2,7)(6,12)(8,14)(13,16)(17,20,23,22)(18,26,28,34)(19,30,33,32)(21,35,40,39)(24,42,44,43)(25,45,46,29)(27,36,47,48)(31,37,38,41)

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=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, #72321763, …
maximal normal subgroup of	#72321740, #72321751, #72321763, …
maximal quotient of	#72343346, #72343383, #72343536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391080);