#202127

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

1016 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)(3,5)(4,11)(6,9)(7,12)(8,10)(13,16)(14,15)(17,18,21,26,22,25,34,39)(19,23,29,37,28,36,43,47)(20,30,32,40,31,24,33,38)(27,35,42,46,41,45,44,48);(1,3,10,9,6,8,5,2)(4,11,12,16,14,15,13,7)(17,19,21,29,22,28,34,43)(18,23,26,37,25,36,39,47)(20,27,33,44,31,41,32,42)(24,35,40,48,30,45,38,46);(1,4,6,14)(2,7,9,16)(3,11,8,15)(5,13,10,12)(17,20)(18,24)(19,27)(21,33)(22,31)(23,35)(25,30)(26,40)(28,41)(29,44)(32,34)(36,45)(37,48)(38,39)(42,43)(46,47);(1,5,6,10)(2,3,9,8)(4,12,14,13)(7,15,16,11)(18,25)(23,36)(24,30)(26,39)(35,45)(37,47)(38,40)(46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,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, 109323);