#240852

order	512
nilpotency class	3
isoclinism	isoclinic class of #240845
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
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

887 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)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,31)(25,38)(30,45)(32,41)(33,40)(35,39)(36,44)(37,43)(42,47)(46,48);(2,7)(3,12)(4,5)(8,15)(9,10)(13,14)(17,19,23,33)(18,24,28,41)(20,35,38,46)(21,32,22,31)(25,42,34,45)(26,29,27,40)(30,37,47,36)(39,44,48,43);(1,3,4,13,6,14,5,12)(2,8,9,16,10,15,7,11)(17,20,21,37,23,38,22,36)(18,25,27,43,28,34,26,44)(19,30,31,46,33,47,32,35)(24,39,29,42,41,48,40,45);(1,4,6,5)(2,7,10,9)(3,13,14,12)(8,11,15,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,40,41,29)(25,43,34,44)(30,35,47,46)(39,42,48,45)

Polycyclic presentation:

2^9|b^2=hi,c^2=e,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dhi,f^c=fhi

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, 148048);