#507936

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 132 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)(3,7)(4,6)(5,8)(17,18)(19,23)(20,24)(21,26)(22,34)(25,31)(27,32)(28,36)(29,39)(30,46)(33,40)(35,42)(37,45)(38,43)(41,47)(44,48);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,29)(21,28)(22,30)(24,39)(25,37)(26,36)(27,38)(31,45)(32,43)(33,42)(34,46)(35,40)(41,48)(44,47);(10,13)(14,16)(18,24)(19,28)(20,31)(22,32)(23,35)(25,27)(26,40)(29,44)(33,47)(34,41)(36,39)(37,46)(38,48)(42,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(19,28)(23,36)(29,42)(30,43)(35,39)(37,48)(38,46)(44,45)

Polycyclic presentation:

2^9|b^2=egi,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415132);