#103116

order	512
nilpotency class	3
isoclinism	isoclinic class of #103025
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

464 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=g,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10312);