#196229

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1083 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 103425);