#202124

order	512
nilpotency class	3
isoclinism	isoclinic class of #201978
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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1062 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109320);