#483963

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
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	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

413 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:

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

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,d^b=dh,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	#72322006, #72322020, #72322029, …
maximal normal subgroup of	#72322006, #72322020, #72322029, …
maximal quotient of	#72343352, #72343388, #72343534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391159);