#194461

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	#60953	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1019 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)(3,9,10,16)(4,8)(5,7)(6,11,14,13)(12,15)(17,18)(19,28,31,40)(20,32)(21,27)(22,36)(23,26)(24,30,39,35)(25,34)(29,44,45,38)(33,41)(37,42)(43,46,48,47);(4,12)(8,15)(9,16)(11,13)(17,19,21,35,23,31,22,30)(18,24,27,40,26,39,36,28)(20,29,34,47,33,45,37,46)(25,38,41,48,42,44,32,43);(1,3)(2,6)(4,13)(5,10)(7,14)(8,16)(9,15)(11,12)(17,20)(18,25)(19,29)(21,34)(22,37)(23,33)(24,38)(26,42)(27,41)(28,43)(30,46)(31,45)(32,36)(35,47)(39,44)(40,48);(2,7)(6,14)(8,15)(9,16)(18,26)(20,33)(24,39)(27,36)(28,40)(29,45)(34,37)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dhi,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, 101657);