#194648

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
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

1092 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101844);