#241068

order	512
nilpotency class	3
isoclinism	isoclinic class of #241060
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

854 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,41)(31,40)(33,44)(35,39)(37,43)(38,42)(45,46)(47,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,23,33)(18,24,28,40)(20,35,38,30)(21,31,22,32)(25,41,43,39)(26,29,27,44)(34,48,42,45)(36,47,37,46);(6,12)(13,16)(17,20,21,37,23,38,22,36)(18,25,27,42,28,43,26,34)(19,30,32,46,33,35,31,47)(24,39,29,45,40,41,44,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,29,40,44)(25,42,43,34)(30,46,35,47)(39,48,41,45)

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,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, 148264);