#194632

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1058 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101828);