#205649

order	512
nilpotency class	3
isoclinism	isoclinic class of #9755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

998 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)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,22,28,23,27,21,26)(19,24,30,38,31,40,36,39)(20,32,34,42,35,41,33,25)(29,43,44,37,45,47,46,48);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,22,36,23,31,21,30)(18,24,26,38,27,40,28,39)(20,29,34,46,35,45,33,44)(25,37,41,47,42,48,32,43);(2,7)(6,14)(8,12)(9,15)(17,20,22,34,23,35,21,33)(18,25,26,41,27,42,28,32)(19,29,36,46,31,45,30,44)(24,37,38,47,40,48,39,43);(17,21,23,22)(18,26,27,28)(19,30,31,36)(20,33,35,34)(24,38,40,39)(25,41,42,32)(29,44,45,46)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112845);