#111204

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18400);