#339893

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247089);