#200731

order	512
nilpotency class	3
isoclinism	isoclinic class of #200728
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1023 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)(3,6)(4,12,15,10)(5,11)(7,9)(8,14,16,13)(17,18)(19,27,30,41)(20,31,33,24)(21,26)(22,25)(23,29,38,34)(28,43)(32,42,35,39)(36,40)(37,44)(45,48)(46,47);(1,3,6,2,7,11,5,9)(4,10,13,8,15,12,14,16)(17,19,21,34,22,30,36,29)(18,23,26,41,25,38,40,27)(20,28,35,44,33,45,32,46)(24,37,42,48,31,47,39,43);(1,4,7,15)(2,8,9,16)(3,10,11,12)(5,14,6,13)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,31)(26,42)(27,43)(29,46)(30,45)(32,36)(34,44)(38,47)(39,40)(41,48);(18,25)(23,38)(24,31)(26,40)(27,41)(37,47)(39,42)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,e^2=hi,f^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=ghi

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, 107927);