#197406

order	512
nilpotency class	3
isoclinism	isoclinic class of #197354
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1036 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dhi,e^c=eh,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, 104602);