#103236

order	512
nilpotency class	3
isoclinism	isoclinic class of #103197
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	C₂×D₄
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

669 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,6)(4,7)(5,8)(9,10,15,20)(11,21,12,23)(13,19,14,18)(16,24,17,22)(25,26,29,32)(27,30,34,37)(28,31,35,39)(33,36,40,38);(2,6)(3,4)(7,8)(9,11,15,12)(10,16,20,17)(13,24,14,22)(18,23,19,21)(25,27,29,34)(26,30,32,37)(28,33,35,40)(31,38,39,36);(2,7)(3,4)(6,8)(9,12,15,11)(10,17,20,16)(13,24,14,22)(18,23,19,21)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^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,d^c=dh,e^c=eg

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, 10432);