#339844

order	512
nilpotency class	3
isoclinism	isoclinic class of #339775
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	C₂³	#537
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

535 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,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=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,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, 247040);