#104051

order	512
nilpotency class	3
isoclinism	isoclinic class of #103989
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

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

428 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11247);