#103013

order	512
nilpotency class	3
isoclinism	isoclinic class of #103005
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

459 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

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, 10209);