#339885

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 247081);