#197012

order	512
nilpotency class	3
isoclinism	isoclinic class of #197011
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	#60546	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1080 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,28)(20,32,35,42)(21,36)(22,26)(23,27)(24,31)(25,34,41,33)(29,44,45,38)(30,40)(37,39)(43,46,48,47);(1,3,4,12,5,11,13,10)(2,6,8,16,7,14,15,9)(17,19,21,31,23,30,22,37)(18,24,26,28,27,39,36,40)(20,29,34,47,35,45,33,46)(25,38,42,43,41,44,32,48);(17,20,23,35)(18,25,27,41)(19,29,30,45)(21,34,22,33)(24,38,39,44)(26,42,36,32)(28,43,40,48)(31,47,37,46);(2,7)(6,14)(8,15)(9,16)(19,30)(24,39)(28,40)(29,45)(31,37)(38,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,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, 104208);