#339772

order	512
nilpotency class	3
isoclinism	isoclinic class of #339703
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

545 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,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=gh,f^c=fi,g^c=gh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246968);