#507786

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	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 592 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,25)(21,24)(22,35)(23,29)(26,34)(27,33)(30,41)(31,38)(32,39)(36,44)(37,48)(40,47)(42,46)(43,45);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,20,29)(18,23,25,28)(21,30,32,44)(22,31,33,45)(24,36,39,41)(26,37,40,42)(27,38,35,43)(34,46,47,48);(11,16)(13,15)(18,24)(21,34)(22,33)(23,36)(25,39)(26,27)(28,41)(30,46)(31,45)(32,47)(35,40)(37,38)(42,43)(44,48);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,25)(19,29)(24,39)(26,40)(27,35)(30,44)(31,45)(46,48)

Polycyclic presentation:

2^9|b^2=i,d^2=egi,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, 414982);