#196322

order	512
nilpotency class	3
isoclinism	isoclinic class of #196320
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1179 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103518);