#332044

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

541 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239240);