#203076

order	512
nilpotency class	3
isoclinism	isoclinic class of #19279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321560, #72322139, #72322149
maximal normal subgroup of	#72321560, #72322139, #72322149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110272);