#204473

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,e^b=eh,g^b=gh,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111669);