#194464

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
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

1012 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)(19,28,30,40)(20,32)(21,26)(22,36)(23,27)(24,31,39,35)(25,34)(29,44,47,38)(33,42)(37,41)(43,45,48,46);(4,12)(8,15)(9,16)(11,13)(17,19,21,35,23,30,22,31)(18,24,26,40,27,39,36,28)(20,29,34,46,33,47,37,45)(25,38,42,48,41,44,32,43);(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)(30,47)(31,45)(32,36)(35,46)(39,44)(40,48);(2,7)(6,14)(8,15)(9,16)(19,30)(20,33)(24,39)(25,41)(28,40)(31,35)(32,42)(34,37)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101660);