#102884

order	512
nilpotency class	3
isoclinism	isoclinic class of #102873
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

649 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,d^c=dh,e^c=eg,f^c=fg

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, 10080);