#197199

order	512
nilpotency class	3
isoclinism	isoclinic class of #197032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1013 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:

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

Polycyclic presentation:

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

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, 104395);