#236073

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6)(7,11)(8,10)(9,13)(12,16)(14,18)(15,17)(19,20)(21,22,26,30)(23,27,32,42)(24,34,35,46)(25,37,40,29)(28,39,44,36)(31,47,48,51)(33,45,38,43)(41,50,52,49);(1,2)(3,4)(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,23)(22,27)(24,31)(25,38)(26,32)(28,41)(29,45)(30,42)(33,40)(34,51)(35,48)(36,50)(37,43)(39,49)(44,52)(46,47);(2,4)(5,7,9,14)(6,11,13,18)(8,15,16,20)(10,17,12,19)(21,24)(22,28)(23,31)(25,39)(26,35)(27,41)(29,46)(30,44)(32,48)(33,50)(34,37)(36,40)(38,49)(42,52)(43,51)(45,47);(7,14)(11,18)(15,20)(17,19)(23,32)(24,35)(27,42)(28,44)(33,38)(34,46)(36,39)(43,45)

Polycyclic presentation:

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

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, 143269);