#204472

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
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×#449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

974 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:

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

Polycyclic presentation:

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

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, 111668);