#236526

order	512
nilpotency class	3
isoclinism	isoclinic class of #236515
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

1132 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,14)(15,16)(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);(1,3,7,15)(2,8,12,4)(5,14,6,9)(10,16,11,13)(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);(1,4)(2,9)(3,10)(5,13)(6,16)(7,8)(11,15)(12,14)(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);(1,5,7,6)(2,10,12,11)(3,14,15,9)(4,13,8,16)(19,28)(23,38)(27,44)(29,34)(37,48)(39,41)(43,47)(45,46)

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,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143722);