#507995

order	512
nilpotency class	4
isoclinism	isoclinic class of #501330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egi,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=cg,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415191);