#236590

order	512
nilpotency class	3
isoclinism	isoclinic class of #236579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1128 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,15)(13,16)(17,18,22,26)(19,23,28,38)(20,30,31,42)(21,33,36,25)(24,35,40,32)(27,43,44,47)(29,41,34,39)(37,46,48,45);(1,3,6,15)(2,7,10,16)(4,13,5,14)(8,11,9,12)(17,19)(18,23)(20,27)(21,34)(22,28)(24,37)(25,41)(26,38)(29,36)(30,47)(31,44)(32,46)(33,39)(35,45)(40,48)(42,43);(1,2)(3,12)(4,9)(5,8)(6,10)(7,14)(11,15)(13,16)(17,20)(18,24)(19,27)(21,35)(22,31)(23,37)(25,42)(26,40)(28,44)(29,46)(30,33)(32,36)(34,45)(38,48)(39,47)(41,43);(1,4,6,5)(2,8,10,9)(3,13,15,14)(7,11,16,12)(19,28)(20,31)(23,38)(24,40)(29,34)(30,42)(32,35)(39,41)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143786);