#194619

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
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

1100 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101815);